Iterative Methods for Solving Nonlinear Equations and Systems

Iterative Methods for Solving N

onlinear Equations and Systems • Juan R. Torregrosa, Alicia Cordero and Fazlollah Soleym

ani

Iterative Methods for Solving Nonlinear Equations and Systems

Printed Edition of the Special Issue Published in Mathematics

www.mdpi.com/journal/mathematics

Juan R. Torregrosa, Alicia Cordero and Fazlollah SoleymaniEdited by

Iterative Methods for SolvingNonlinear Equations and Systems

Iterative Methods for SolvingNonlinear Equations and Systems

Special Issue Editors

Juan R. Torregrosa

Alicia Cordero

Fazlollah Soleymani

MDPI • Basel • Beijing •Wuhan • Barcelona • Belgrade


Juan R. Torregrosa

Polytechnic University of

Valencia

Spain

Alicia Cordero

Polytechnic University of

Valencia

Spain

Fazlollah Soleymani

Institute for Advanced Studies in

Basic Sciences (IASBS)

Iran

Editorial Office

MDPI

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4052 Basel, Switzerland

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Contents

About the Special Issue Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Preface to ”Iterative Methods for Solving Nonlinear Equations and Systems” . . . . . . . . . . xi

Shengfeng Li, Xiaobin Liu and Xiaofang Zhang

A Few Iterative Methods by Using [1, n]-Order Pade Approximation of Function and theImprovementsReprinted from: Mathematics 2019, 7, 55, doi:10.3390/math7010055 . . . . . . . . . . . . . . . . . . 1

Fuad W. Khdhr, Rostam K. Saeed and Fazlollah Soleymani

Improving the Computational Efficiency of a Variant of Steffensen’s Method for NonlinearEquationsReprinted from: Mathematics 2019, 7, 306, doi:10.3390/math7030306 . . . . . . . . . . . . . . . . . 15

Yanlin Tao and Kalyanasundaram Madhu

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniquesand Their Basins of Attraction and Its ApplicationReprinted from: Mathematics 2019, 7, 322, doi:10.3390/math7040322 . . . . . . . . . . . . . . . . . 24

Prem B. Chand, Francisco I. Chicharro, Neus Garrido and Pankaj Jain

Design and Complex Dynamics of Potra–Ptak-Type Optimal Methods for Solving NonlinearEquations and Its ApplicationsReprinted from: Mathematics 2019, 7, 942, doi:10.3390/math7100942 . . . . . . . . . . . . . . . . . 46

Jian Li, Xiaomeng Wang and Kalyanasundaram Madhu

Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and TheirBasins of AttractionReprinted from: Mathematics 2019, 7, 1052, doi:10.3390/math7111052 . . . . . . . . . . . . . . . . 67

Min-Young Lee, Young Ik Kim and Beny Neta

A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their DynamicsUnderlying Purely Imaginary Extraneous Fixed PointsReprinted from: Mathematics 2019, 7, 562, doi:10.3390/math7060562 . . . . . . . . . . . . . . . . . 82

Fiza Zafar, Alicia Cordero and Juan R. Torregrosa

An Efficient Family of Optimal Eighth-Order Multiple Root FindersReprinted from: Mathematics 2018, 6, 310, doi:10.3390/math6120310 . . . . . . . . . . . . . . . . . 108

Ramandeep Behl, Eulalia Martınez, Fabricio Cevallos and Diego Alarcon

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple RootsReprinted from: Mathematics 2019, 7, 55, doi:10.3390/math7040339 . . . . . . . . . . . . . . . . . . 124

Saima Akram, Fiza Zafar and Nusrat Yasmin

An Optimal Eighth-Order Family of Iterative Methods for Multiple RootsReprinted from: Mathematics 2019, 7, 672, doi:10.3390/math7080672 . . . . . . . . . . . . . . . . . 135

Auwal Bala Abubakar, Poom Kumam, Hassan Mohammad and Aliyu Muhammed Awwal

An Efficient Conjugate Gradient Method for Convex Constrained Monotone NonlinearEquations with ApplicationsReprinted from: Mathematics 2019, 7, 767, doi:10.3390/math7090767 . . . . . . . . . . . . . . . . . 149

v

Ioannis K. Argyros, A. Alberto Magrenan, Lara Orcos and Inigo Sarrıa

Advances in the Semilocal Convergence of Newton’s Method with Real-World ApplicationsReprinted from: Mathematics 2019, 7, 299, doi:10.3390/math7030299 . . . . . . . . . . . . . . . . . 174

D. R. Sahu, Ravi P. Agarwal and Vipin Kumar Singh

A Third Order Newton-Like Method and Its ApplicationsReprinted from: Mathematics 2019, 7, 31, doi:10.3390/math7010031 . . . . . . . . . . . . . . . . . . 186

Ioannis K. Argyros and Ramandeep Behl

Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak ConditionsReprinted from: Mathematics 2019, 7, 89, doi:10.3390/math7010089 . . . . . . . . . . . . . . . . . . 208

Ioannis K. Argyros and Stepan Shakhno

Extended Local Convergence for the Combined Newton-Kurchatov Method Under theGeneralized Lipschitz ConditionsReprinted from: Mathematics 2019, 7, 207, doi:10.3390/math7020207 . . . . . . . . . . . . . . . . . 222

Cristina Amoros, Ioannis K. Argyros, Ruben Gonzalez

Study of a High Order Family: Local Convergence and DynamicsReprinted from: Mathematics 2019, 7, 225, doi:10.3390/math7030225 . . . . . . . . . . . . . . . . . 234

Zhang Yong, Neha Gupta, J. P. Jaiswal and Kalyanasundaram Madhu

On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded ThirdDerivative CaseReprinted from: Mathematics 2019, 7, 540, doi:10.3390/math7060540 . . . . . . . . . . . . . . . . . 248

Pawicha Phairatchatniyom, Poom Kumam, Yeol Je Cho, Wachirapong Jirakitpuwapat and

Kanokwan Sitthithakerngkiet

The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem forMulti-Valued Quasi Nonexpansive Mappings with Some ApplicationsReprinted from: Mathematics 2019, 7, 560, doi:10.3390/math7060560 . . . . . . . . . . . . . . . . . 262

Mozafar Rostami, Taher Lotfi and Ali Brahmand

A Fast Derivative-Free Iteration Scheme for Nonlinear Systems and Integral EquationsReprinted from: Mathematics 2019, 7, 637, doi:10.3390/math7070637 . . . . . . . . . . . . . . . . . 284

Sergio Amat, Ioannis Argyros, Sonia Busquier, Miguel Angel Hernandez-Veron and

Marıa Jesus Rubio

A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth OperatorsReprinted from: Mathematics 2019, 7, 701, doi:10.3390/math7080701 . . . . . . . . . . . . . . . . . 295

Auwal Bala Abubakar, Poom Kumam, Hassan Mohammad, Aliyu Muhammed Awwal and

Kanokwan Sitthithakerngkiet

A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equationswith Some ApplicationsReprinted from: Mathematics 2019, 7, 745, doi:10.3390/math7080745 . . . . . . . . . . . . . . . . . 307

Alicia Cordero, Cristina Jordan, Esther Sanabria and Juan R. Torregrosa

A New Class of Iterative Processes for Solving Nonlinear Systems by Using One DividedDifferences OperatorReprinted from: Mathematics 2019, 7, 776, doi:10.3390/math7090776 . . . . . . . . . . . . . . . . . 332

vi

Abdolreza Amiri, Mohammad Taghi Darvishi, Alicia Cordero and Juan Ramon Torregrosa

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly NonlinearSystemsReprinted from: Mathematics 2019, 7, 815, doi:10.3390/math7090815 . . . . . . . . . . . . . . . . . 344

Hessah Faihan Alqahtani, Ramandeep Behl, Munish Kansal

Higher-Order Iteration Schemes for Solving Nonlinear Systems of EquationsReprinted from: Mathematics 2019, 7, 937, doi:10.3390/math7100937 . . . . . . . . . . . . . . . . . 361

Dilan Ahmed, Mudhafar Hama, Karwan Hama Faraj Jwamer and Stanford Shateyi

A Seventh-Order Scheme for Computing the Generalized Drazin InverseReprinted from: Mathematics 2019, 7, 622, doi:10.3390/math7070622 . . . . . . . . . . . . . . . . . 375

Haifa Bin Jebreen

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration SchemeReprinted from: Mathematics 2019, 7, 731, doi:10.3390/math7080731 . . . . . . . . . . . . . . . . . 385

Zhinan Wu and Xiaowu Li

An Improved Curvature Circle Algorithm for Orthogonal Projection onto a Planar AlgebraicCurveReprinted from: Mathematics 2019, 7, 912, doi:10.3390/math7100912 . . . . . . . . . . . . . . . . . 396

Anantachai Padcharoen and Pakeeta Sukprasert

Nonlinear Operators as Concerns Convex Programming and Applied to Signal ProcessingReprinted from: Mathematics 2019, 7, 866, doi:10.3390/math7090866 . . . . . . . . . . . . . . . . . 420

Juan Liang, Linke Hou, Xiaowu Li, Feng Pan, Taixia Cheng and Lin Wang

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve inn-Dimensional Euclidean SpaceReprinted from: Mathematics 2018, 6, 306, doi:10.3390/math6120306 . . . . . . . . . . . . . . . . . 435

Jose Antonio Ezquerro and Miguel Angel Hernandez–Veron

How to Obtain Global Convergence Domains via Newton’s Method for Nonlinear IntegralEquationsReprinted from: Mathematics 2018, 6, 553, doi:10.3390/math7060553 . . . . . . . . . . . . . . . . . 458

Malik Zaka Ullah

Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FDSchemeReprinted from: Mathematics 2019, 7, 704, doi:10.3390/math7080704 . . . . . . . . . . . . . . . . . 467

vii

About the Special Issue Editors

Juan Ramon Torregrosa, Dr., has a Bachelor in Mathematical Sciences (Universitat de Valencia)

and obtained his PhD (1990, Universitat de Valencia) defending his thesis “Algunas propiedades

geometricas uniformes y no uniformes de un espacio de Banach.” He is Full Professor of Applied

Mathematics in the Institute for Multidisciplinary Mathematics of the Polytechnical University of

Valencia.

He published several papers about locally convex spaces and Banach spaces in the 1990s.

Afterwards, he launched new research projects in linear algebra, matrix analysis, and combinatorics.

He has supervised several PhD theses on these topics. He also has published a significant number of

papers in related journals: Linear Algebra and Its Applications, Applied Mathematics Letters, and SIAM

Journal Matrix Analysis.

His current research is in numerical analysis. He focuses on different problems related to the

solution of nonlinear equations and systems, matrix equations, and dynamical analysis of rational

functions involved in iterative methods. He has published more than 200 papers in JCR journals and

he also has presented numerous communications in international conferences.

Alicia Cordero, Dr., has a Bachelor in Mathematic Sciences (1995, Universitat de Valencia). She

obtained her PhD in Mathematics (2003, Universitat Jaume I) defending her PhD Thesis “Cadenas

de orbitas periodicas en la variedad S2xS1,“ which was supervised by Jose Martınez and Pura Vindel.

Through the years, she has published many papers about the decomposition into round loops

of three-dimensional varieties, links of periodic orbits of non-singular Morse-Smale fluxes, and their

applications to celestial mechanics.

She is Full Professor of Applied Mathematics in the Institute for Multidisciplinary Mathematics

of the Polytechnical University of Valencia. Her current research is focused on dynamical systems

and numerical analysis, highlighting the iterative methods for solving nonlinear equations and

systems as well as the dynamical study of the rational functions involved in these processes. She has

published more than 150 papers in JCR Journals. She also has presented numerous communications

in international conferences.

Fazlollah Soleymani, Dr., acquired his Ph.D. degree in numerical analysis at Ferdowsi University of

Mashhad in Iran. He also accomplished his postdoctoral fellowship at the Polytechnic University of

Valencia in Spain. Soleymani’s main research interests are in the areas of computational mathematics.

Recently, he has been working on a number of different problems, which fall under semi-discretized

and (localized) RBF(–(H)FD) meshfree schemes for financial partial (integro–) differential equations,

high-order iterative methods for nonlinear systems, numerical solution of stochastic differential

equations, and iteration methods for generalized inverses.

ix

Preface to ”Iterative Methods for Solving

Nonlinear Equations and Systems”

Solving nonlinear equations in any Banach space (real or complex nonlinear equations, nonlinear

systems, and nonlinear matrix equations, among others) is a non-trivial task that involves many areas

of science and technology. Usually the solution is not directly affordable and requires an approach

utilizing iterative algorithms. This is an area of research that has grown exponentially over the last

few years.

This Special Issue focuses mainly on the design, analysis of convergence, and stability of

new iterative schemes for solving nonlinear problems and their application to practical problems.

Included papers study the following topics: Methods for finding simple or multiple roots, either

with or without derivatives, iterative methods for approximating different generalized inverses, and

real or complex dynamics associated with the rational functions resulting from the application of

an iterative method on a polynomial function. Additionally, the analysis of the convergence of the

proposed methods has been carried out by means of different sufficient conditions assuring the local,

semilocal, or global convergence.

This Special issue has allowed us to present the latest research results in the area of iterative

processes for solving nonlinear equations as well as systems and matrix equations. In addition to

the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, partial

differential equations, or convex programming reveal the connection between iterative methods and

other branches of science and engineering.

Juan R. Torregrosa, Alicia Cordero, Fazlollah Soleymani


xi

mathematics

Article

A Few Iterative Methods by Using [1, n]-Order PadéApproximation of Function and the Improvements

Shengfeng Li 1,* , Xiaobin Liu 2 and Xiaofang Zhang 1

1 Institute of Applied Mathematics, Bengbu University, Bengbu 233030, China; [email protected] School of Computer Engineering, Bengbu University, Bengbu 233030, China; [email protected]* Correspondence: [email protected]; Tel.: +86-552-317-5158

Received: 15 November 2018; Accepted: 30 December 2018; Published: 7 January 2019

Abstract: In this paper, a few single-step iterative methods, including classical Newton’s methodand Halley’s method, are suggested by applying [1, n]-order Padé approximation of function forfinding the roots of nonlinear equations at first. In order to avoid the operation of high-orderderivatives of function, we modify the presented methods with fourth-order convergence by usingthe approximants of the second derivative and third derivative, respectively. Thus, several modifiedtwo-step iterative methods are obtained for solving nonlinear equations, and the convergence of thevariants is then analyzed that they are of the fourth-order convergence. Finally, numerical experimentsare given to illustrate the practicability of the suggested variants. Henceforth, the variants withfourth-order convergence have been considered as the imperative improvements to find the roots ofnonlinear equations.

Keywords: nonlinear equations; Padé approximation; iterative method; order of convergence;numerical experiment

1. Introduction

It is well known that a variety of problems in different fields of science and engineering requireto find the solution of the nonlinear equation f (x) = 0 where f : I → D, for an interval I ⊆ R andD ⊆ R, is a scalar function. In general, iterative methods, such as Newton’s method, Halley’s method,Cauchy’s method, and so on, are the most used techniques. Hence, iterative algorithms based on theseiterative methods for finding the roots of nonlinear equations are becoming one of the most importantaspects in current researches. We can see the works, for example, [1–22] and references therein. In thelast few years, some iterative methods with high-order convergence have been introduced to solvea single nonlinear equation. By using various techniques, such as Taylor series, quadrature formulae,decomposition techniques, continued fraction, Padé approximation, homotopy methods, Hermiteinterpolation, and clipping techniques, these iterative methods can be constructed. For instance,there are many ways of introducing Newton’s method. Among these ways, using Taylor polynomials toderive Newton’s method is probably the most widely known technique [1,2]. By considering differentquadrature formulae for the computation of the integral, Weerakoon and Fernando derive an implicititerative scheme with cubic convergence by the trapezoidal quadrature formulae [4], while Corderoand Torregrosa develope some variants of Newton’s method based in rules of quadrature of fifthorder [5]. In 2005, Chun [6] have presented a sequence of iterative methods improving Newton’smethod for solving nonlinear equations by applying the Adomian decomposition method. Basedon Thiele’s continued fraction of the function, Li et al. [7] give a fourth-order convergent iterativemethod. Using Padé approximation of the function, Li et al. [8] rederive the Halley’s method andby the divided differences to approximate the derivatives, they arrive at some modifications withthird-order convergence. In [9], Abbasbandy et al. present an efficient numerical algorithm for

Mathematics 2019, 7, 55; doi:10.3390/math7010055 www.mdpi.com/journal/mathematics1

Mathematics 2019, 7, 55

solving nonlinear algebraic equations based on Newton–Raphson method and homotopy analysismethod. Noor and Khan suggest and analyze a new class of iterative methods by using the homotopyperturbation method in [10]. In 2015, Wang et al. [11] deduce a general family of n-point Newton typeiterative methods for solving nonlinear equations by using direct Hermite interpolation. Moreover, fora particular class of functions, for instance, if f is a polynomial, there exist some efficient univariateroot-finding algorithms to compute all solutions of the polynomial equation (see [12,13]). In theliterature [13], Barton et al. present an algorithm for computing all roots of univariate polynomialbased on degree reduction, which has the higher convergence rate than Newton’s method. In thisarticle, we will mainly solve more general nonlinear algebraic equations.

Newton’s method is probably the best known and most widely used iterative algorithm forroot-finding problems. By applying Taylor’s formula for the function f (x), let us recall briefly how toderive Newton iterative method. Suppose that f (x) ∈ Cn[I], n = 1, 2, 3, . . ., and η ∈ I is a single rootof the nonlinear equation f (x) = 0. For a given guess value x0 ∈ I and a δ ∈ R, assume that f ′(x) �= 0for each x belongs to the neighborhood (x0 − δ, x0 + δ). For any x ∈ (x0 − δ, x0 + δ), we expand f (x)into the following Taylor’s formula about x0:

f (x) = f (x0) + f ′(x0)(x− x0) +12!

f ′′(x0)(x− x0)2 + · · ·+ 1

k!(x− x0)

k f (k)(x0) + · · · ,

where k = 0, 1, 2, · · · . Let |η− x0| be sufficiently small. Then the terms involving (η− x0)k, k = 2, 3, . . . ,

are much smaller. Hence, we think the fact that the first Taylor polynomial is a good approximation tothe function near the point x0 and give that

f (x0) + f ′(x0)(η − x0) ≈ 0.

Notice the fact f ′(x0) �= 0, and solving the above equation for η yields

η ≈ x0 − f (x0)

f ′(x0),

which follows that we can construct the Newton iterative scheme as below

xk+1 = xk − f (xk)

f ′(xk), k = 0, 1, 2, . . . .

It has been known that Newton iterative method is a celebrated one-step iterative method.The order of convergence of Newton’s method is quadratic for a simple zero and linear for multiple root.

Motivated by the idea of the above technique, in this paper, we start with using Padéapproximation of a function to construct a few one-step iterative schemes which includes classicalNewton’s method and Halley’s method to find roots of nonlinear equations. In order to avoidcalculating the high-order derivatives of the function, then we employ the approximants of the higherderivatives to improve the presented iterative method. As a result, we build several two-step iterativeformulae, and some of them do not require the operation of high-order derivatives. Furthermore, it isshown that these modified iterative methods are all fouth-order convergent for a simple root of theequation. Finally, we give some numerical experiments and comparison to illustrate the efficiency andperformance of the presented methods.

The rest of this paper is organized as follows. we introduce some basic preliminaries about Padéapproximation and iteration theory for root-finding problem in Section 2. In Section 3, we firstlyconstruct several one-step iterative schemes based on Padé approximation. Then, we modify thepresented iterative method to obtain a few iterative formulae without calculating the high-orderderivatives. In Section 4, we show that the modified methods have fourth-order convergence at leastfor a simple root of the equation. In Section 5 we give numerical examples to show the performance of

2


the presented methods and compare them with other high-order methods. At last, we draw conclusionsfrom the experiment results in Section 6.

2. Preliminaries

In this section, we briefly review some basic definitions and results for Padé approximation offunction and iteration theory for root-finding problem. Some surveys and complete literatures aboutiteration theory and Padé approximation could be found in Alfio [1], Burden et al. [2], Wuytack [23],and Xu et al. [24].

Definition 1. Assume that f (x) is a function whose (n + 1)-st derivative f (n+1)(x), n = 0, 1, 2, . . . , existsfor any x in an interval I. Then for each x ∈ I, we have

f (x) = f (x0) + f ′(x0)(x− x0) +f ′′(x0)

2!(x− x0)

2 + · · ·+ f (n)(x0)

n!(x− x0)

n + o[(x− x0)n], (1)

which is called the Taylor’s formula with Peano remainder term of order n based at x0, and the error o[(x− x0)n]

is called the Peano remainder term or the Peano truncation error.

Definition 2. If P(x) is a polynomial, the accurate degree of the polynomial is ∂(P), and the order of thepolynomial is ω(P), which is the degree of the first non-zero term of the polynomial.

Definition 3. If it can be found two ploynomials

P(x) =m

∑i=0

ai(x− x0)i and Q(x) =

n

∑i=0

bi(x− x0)i

such that∂(P(x)) ≤ m, ∂(Q(x)) ≤ n, ω( f (x)Q(x)− P(x)) ≥ m + n + 1,

then we have the following incommensurable form of the rational fraction P(x)Q(x) :

Rm,n(x) =P0(x)Q0(x)

=P(x)Q(x)

,

which is called [m, n]-order Padé approximation of function f (x).

We give the computational formula of Padé approximation of function f (x) by the use ofdeterminant, as shown in the following lemma [23,24].

Lemma 1. Assume that Rm,n(x) = P0(x)Q0(x) is Padé approximation of function f (x). If the matrix

Am,n =

⎛⎜⎜⎜⎜⎝am am−1 · · · am+1−n

am+1 am · · · am+2−n...

.... . .

...am+n−1 am+n−2 · · · am

⎞⎟⎟⎟⎟⎠is nonsingular, that is the determinant |Am,n| = d �= 0, then P0(x), Q0(x) can be written by the followingdeterminants

3


P0(x) =1d

∣∣∣∣∣∣∣∣∣∣Tm(x) (x− x0)Tm−1(x) · · · (x− x0)

nTm+1−n(x)am+1 am · · · am+1−n

......

. . ....

am+n am+n−1 · · · am

∣∣∣∣∣∣∣∣∣∣and

Q0(x) =1d

∣∣∣∣∣∣∣∣∣∣1 (x− x0) · · · (x− x0)

n

am+1 am · · · am+1−n...

.... . .

...am+n am+n−1 · · · am

∣∣∣∣∣∣∣∣∣∣,

where an = f (n)(x0)n! , n = 0, 1, 2, . . ., and we appoint that

Tk(x) =

⎧⎨⎩k∑

i=0ai(x− x0)

i, f or k ≥ 0,

0, f or k < 0.

Next, we recall the speed of convergence of an iterative scheme. Thus, we give the followingdefinition and lemma.

Definition 4. Assume that a sequence {xi}∞i=0 converges to η, with xi �= η for all i, i = 0, 1, 2, . . .. Let the

error be ei = xi − η. If there exist two positive constants α and β such that

limi→∞

|ei+1||ei|α = β,

then {xi}∞i=0 converges to the constant η of order α. When α = 1, the sequence {xi}∞

i=0 is linearly convergent.When α > 1, the sequence {xi}∞

i=0 is said to be of higher-order convergence.

For a single-step iterative method, sometimes it is convenient to use the following lemma to judgethe order of convergence of the iterative method.

Lemma 2. Assume that the equation f (x) = 0, x ∈ I, can be rewritten as x = ϕ(x), where f (x) ∈ C[I] andϕ(x) ∈ Cγ[I], γ ∈ N+. Let η be a root of the equation f (x) = 0. If the iterative function ϕ(x) satisfies

ϕ(j)(η) = 0, j = 1, 2, . . . , γ− 1, ϕ(γ)(η) �= 0,

then the order of convergence of the iterative scheme xi+1 = ϕ(xi), i = 0, 1, 2, . . ., is γ.

3. Some Iterative Methods

Let η be a simple real root of the equation f (x) = 0, where f : I → D, I ⊆ R, D ⊆ R. Supposethat x0 ∈ I is an initial guess value sufficiently close to η, and the function f (x) has n-th derivativef (n)(x), n = 1, 2, 3, . . . , in the interval I. According to Lemma 1, [m, n]-order Padé approximation offunction f (x) is denoted by the following rational fraction:

4


f (x) ≈ Rm,n(x) =

∣∣∣∣∣∣∣∣∣∣Tm(x) (x− x0)Tm−1(x) · · · (x− x0)

nTm+1−n(x)am+1 am · · · am+1−n

......

. . ....

am+n am+n−1 · · · am

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣1 (x− x0) · · · (x− x0)

n

am+1 am · · · am+1−n...

.... . .

...am+n am+n−1 · · · am

∣∣∣∣∣∣∣∣∣∣

. (2)

Recall Newton iterative method derived by Taylor’s series in Section 1. The first Taylor polynomialis regarded as a good approximation to the function f (x) near the point x0. Solving the linear equationdenoted by f (x0) + f ′(x0)(η − x0) ≈ 0 for η gives us the stage for Newton’s method. Then, we thinkwhether or not a novel or better linear function is selected to approximate the function f (x) near thepoint x0. Maybe Padé approximation can solve this question. In the process of obtaining new iterativemethods based on Padé approximation of function, on the one hand, we consider that the degree of thenumerator of Equation (2) is always taken as 1, which guarantees to obtain the different linear function.On the other hand, we discuss the equations are mainly nonlinear algebraic equations, which differrational equations and have not the poles. Clearly, as n grows, the poles of the denominator ofEquation (2) do not affect the linear functions that we need. These novel linear functions may be ableto set the stage for new methods. Next, let us start to introduce a few iterative methods by using[1, n]-order Padé approximation of function.

3.1. Iterative Method Based on [1, 0]-Order Padé Approximation

Firstly, when m = 1, n = 0, we consider [1, 0]-order Padé approximation of function f (x).It follows from the expression (2) that

f (x) ≈ R1,0(x) = T1(x) = a0 + a1(x− x0).

Let R1,0(x) = 0, then we havea0 + a1(x− x0) = 0. (3)

Due to the determinant |A1,0| �= 0, i.e., f ′(x0) �= 0, we obtain the following equation fromEquation (3).

x = x0 − a0

a1.

In view of a0 = f (x0), a1 = f ′(x0), we reconstruct the Newton iterative method as below.

Method 1. Assume that the function f : I → D has its first derivative at the point x0 ∈ I. Then we obtain thefollowing iterative method based on [1, 0]-order Padé approximation of function f (x):

xk+1 = xk − f (xk)

f ′(xk), k = 0, 1, 2, . . . . (4)

Starting with an initial approximation x0 that is sufficiently close to the root η and using the abovescheme (4), we can get the iterative sequence {xi}∞

i=0.

Remark 1. Method 1 is the well-known Newton’s method for solving nonlinear equation [1,2].

5



Secondly, when m = 1, n = 1, we think about [1, 1]-order Padé approximation of function f (x).Similarly, it follows from the expression (2) that

f (x) ≈ R1,1(x) =

∣∣∣∣∣ T1(x) (x− x0)T0(x)a2 a1

∣∣∣∣∣∣∣∣∣∣ 1 (x− x0)

a2 a1

∣∣∣∣∣.

Let R1,1(x) = 0, then we get

a0a1 + a21(x− x0)− a0a2(x− x0) = 0. (5)

Due to the determinant |A1,1| �= 0, that is,∣∣∣∣∣ a1 a0

a2 a1

∣∣∣∣∣ =∣∣∣∣∣ f ′(x0) f (x0)

f ′′(x0)2 f ′(x0)

∣∣∣∣∣ = f ′2(x0)− f (x0)f ′′(x0)

2�= 0.

Thus, we obtain the following equality from Equation (5):

x = x0 − a0a1

a21 − a0a2

.

Combining a0 = f (x0), a1 = f ′(x0), a2 = 12 f ′′(x0), gives Halley iterative method as follows.

Method 2. Assume that the function f : I → D has its second derivative at the point x0 ∈ I. Then we obtainthe following iterative method based on [1, 1]-order Padé approximation of function f (x):

xk+1 = xk − 2 f (xk) f ′(xk)

2 f ′2(xk)− f (xk) f ′′(xk), k = 0, 1, 2, . . . . (6)

Starting with an initial approximation x0 that is sufficiently close to the root η and applying the abovescheme (6), we can obtain the iterative sequence {xi}∞

i=0.

Remark 2. Method 2 is the classical Halley’s method for finding roots of nonlinear equation [1,2],which converges cubically.


Thirdly, when m = 1, n = 2, we take into account [1, 2]-order Padé approximation of functionf (x). By the same manner, it follows from the expression (2) that

f (x) ≈ R1,2(x) =

∣∣∣∣∣∣∣T1(x) (x− x0)T0(x) 0

a2 a1 a0

a3 a2 a1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣1 x− x0 (x− x0)

2

a2 a1 a0

a3 a2 a1

∣∣∣∣∣∣∣.

Let R1,2(x) = 0, then one has

a0a21 + a2

0a2 + (a31 − 2a0a1a2 + a2

0a3)(x− x0) = 0. (7)

6


Due to the determinant |A1,2| �= 0, that is,∣∣∣∣∣∣∣a1 a0 0a2 a1 a0

a3 a2 a1

∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣

f ′(x0) f (x0) 0f ′′(x0)

2 f ′(x0) f (x0)f ′′′(x0)

6f ′′(x0)

2 f ′(x0)

∣∣∣∣∣∣∣ = f ′3(x0)− f (x0) f ′(x0) f ′′(x0) +f 2(x0) f ′′′(x0)

6�= 0.

Thus, we gain the following equality from Equation (7):

x = x0 − a0a21 − a2

0a2

a31 − 2a0a1a2 + a2

0a3.

Substituting a0 = f (x0), a1 = f ′(x0), a2 = 12 f ′′(x0), and a3 = 1

6 f ′′′(x0) into the above equationgives a single-step iterative method as follows.

Method 3. Assume that the function f : I → D has its third derivative at the point x0 ∈ I. Then we obtainthe following iterative method based on [1, 2]-order Padé approximation of function f (x):

xk+1 = xk −3 f (xk)

(2 f ′2(xk)− f (xk) f ′′(xk)

)6 f ′3(xk)− 6 f (xk) f ′(xk) f ′′(xk) + f 2(xk) f ′′′(xk)

, k = 0, 1, 2, . . . . (8)

Starting with an initial approximation x0 that is sufficiently close to the root η and applying the abovescheme (8), we can receive the iterative sequence {xi}∞

i=0.

Remark 3. Method 3 could be used to find roots of nonlinear equation. Clearly, for the sake of applying thisiterative method, we must compute the second derivative and the third derivative of the function f (x), which maygenerate inconvenience. In order to overcome the drawback, we suggest approximants of the second derivativeand the third derivative, which is a very important idea and plays a significant part in developing some iterativemethods free from calculating the higher derivatives.

3.4. Modified Iterative Method Based on Approximant of the Third Derivative

In fact, we let zk = xk − f (xk)f ′(xk)

. Then expanding f (zk) into third Taylor’s series about the pointxk yields

f (zk) ≈ f (xk) + f ′(xk)(zk − xk) +12!

f ′′(xk)(zk − xk)2 +

13!

f ′′′(xk)(zk − xk)3,

which follows that

f ′′′(xk) ≈ 3 f 2(xk) f ′(xk) f ′′(xk)− 6 f (zk) f ′3(xk)

f 3(xk). (9)

Substituting (9) into (8), we can have the following iterative method.

Method 4. Assume that the function f : I → D has its second derivative about the point x0 ∈ I. Then wepossess a modified iterative method as below:⎧⎨⎩ zk = xk − f (xk)

f ′(xk),

xk+1 = xk − xk−zk1+2 f (zk) f ′2(xk)L−1(xk)

, k = 0, 1, 2, . . . ,(10)

where L(xk) = f (xk)(

f (xk) f ′′(xk)− 2 f ′2(xk)). Starting with an initial approximation x0 that is sufficiently

close to the root η and using the above scheme (10), we can have the iterative sequence {xi}∞i=0.

Remark 4. Methods 4 is a two-step iterative method free from third derivative of the function.

7


3.5. Modified Iterative Method Based on Approximant of the Second Derivative

It is obvious that the iterative method (10) requires the operation of the second derivative of thefunction f (x). In order to avoid computing the second derivative, we introduce an approximant of thesecond derivative by using Taylor’s series.

Similarly, expanding f (zk) into second Taylor’s series about the point xk yields

f (zk) ≈ f (xk) + (zk − xk) f ′(xk) +12!(zk − xk)

2 f ′′(xk),

which means

f ′′(xk) ≈ 2 f (zk) f ′2(xk)

f 2(xk). (11)

Using (11) in (10), we can get the following modified iterative method without computing secondderivative.

Method 5. Assume that the function f : I → D has its first derivative about the point x0 ∈ I. Then we havea modified iterative method as below:⎧⎨⎩ zk = xk − f (xk)

f ′(xk),

xk+1 = xk − f (xk)− f (zk)f (xk)−2 f (zk)

(xk − zk), k = 0, 1, 2, . . . .(12)

Starting with an initial approximation x0 that is sufficiently close to the root η and using the abovescheme (12), we can obtain the iterative sequence {xi}∞

i=0.

Remark 5. Method 5 is another two-step iterative method. It is clear that Method 5 does not require to calculatethe high-order derivative. But more importantly, the characteristic of Method 5 is that per iteration it requirestwo evaluations of the function and one of its first-order derivative. The efficiency of this method is better thanthat of the well-known other methods involving the second-order derivative of the function.

4. Convergence Analysis of Iterative Methods

Theorem 1. Suppose that f (x) is a function whose n-th derivative f (n)(x), n = 1, 2, 3, . . ., exists ina neighborhood of its root η with f ′(η) �= 0. If the initial approximation x0 is sufficiently close to η, then theMethod 3 defined by (8) is fourth-order convergent.

Proof of Theorem 1. By the hypothesis f (η) = 0 and f ′(η) �= 0, we know that η is an unique singleroot of the equation f (x) = 0. So, for each positive integer n ≥ 1, we have that the derivatives of highorders f (n)(η) �= 0. Considering the iterative scheme (8) in Method 3, we denote its correspondingiterative function as shown below:

ϕ(x) = x− 3 f (x)(2 f ′2(x)− f (x) f ′′(x)

)6 f ′3(x)− 6 f (x) f ′(x) f ′′(x) + f 2(x) f ′′′(x)

. (13)

By calculating the first and high-order derivatives of the iterative function ϕ(x) with respect to xat the point η, we verify that

ϕ′(η) = 0, ϕ′′(η) = 0, ϕ′′′(η) = 0

and

ϕ(4)(η) =3 f ′′3(η)− 4 f ′(η) f ′′(η) f ′′′(η) + f ′2(η) f (4)(η)

f ′3(η)�= 0.

8


Thus, it follows from Lemma 2 that Method 3 defined by (8) is fourth-order convergent.This completes the proof.

Theorem 2. Suppose that f (x) is a function whose n-th derivative f (n)(x), n = 1, 2, 3, . . ., exists ina neighborhood of its root η with f ′(η) �= 0. If the initial approximation x0 is sufficiently close to η, then theMethod 4 defined by (10) is at least fourth-order convergent with the following error equation

ek+1 = (b32 − 2b2b3)e4

k + O(e5k),

where ek = xk − η, k = 1, 2, 3, . . ., and the constants bn = anf ′(η) , an = f (n)(η)

n! , n = 1, 2, 3, . . ..

Proof of Theorem 2. By the hypothesis, it is clear to see that η is an unique single root of the equationf (x) = 0. By expanding f (xk), f ′(xk) and f ′′(xk) into Taylor’s series about η, we obtain

f (xk) = ek f ′(η) + e2k

2! f ′′(η) + e3k

3! f ′′′(η) + e4k

4! f (4)(η) + e5k

5! f (5)(η) + e6k

6! f (6)(η) + O(e7k)

= f ′(η)(b1ek + b2e2

k + b3e3k + b4e4

k + b5e5k + b6e6

k + O(e7k))

,(14)

f ′(xk) = f ′(η)(

b1 + 2b2ek + 3b3e2k + 4b4e3

k + 5b5e4k + 6b6e5

k + O(e6k))

(15)

andf ′′(xk) = f ′(η)

(2b2 + 6b3ek + 12b4e2

k + 20b5e3k + 30b6e4

k + O(e5k))

, (16)

where bn = 1n!

f (n)(η)f ′(η) , n = 1, 2, · · · . Clearly, b1 = 1. Dividing (14) by (15) directly, gives us

f (xk)f ′(xk)

= xk − zk = ek − b2e2k − 2(b3 − b2

2)e3k − (4b3

2 + 3b4 − 7b2b3)e4k

−2(10b22b3 − 2b5 + 5b2b4 + 4b4

2 + 3b23)e

5k

−(16b52 + 28b2

2b4 + 33b2b23 + 5b6 − 52b3

2b3 − 17b3b4 − 13b2b5)e6k + O(e7

k).

(17)

By substituting (17) into (10) in Method 4, one has

zk = η + b2e2k + 2(b3 − b2

2)e3k + (4b3

2 + 3b4 − 7b2b3)e4k

+2(10b22b3 − 2b5 + 5b2b4 + 4b4

2 + 3b23)e

5k

+(16b52 + 28b2

2b4 + 33b2b23 + 5b6 − 52b3

2b3 − 17b3b4 − 13b2b5)e6k + O(e7

k).

(18)

Again, expanding f (zk) by Taylor’s series about η, we have

f (zk) = f ′(η)(b2e2

k − 2(b22 − b3)e3

k − (7b2b3 − 5b32 − 3b4)e4

k

−2(5b2b4 + 6b42 + 3b2

3 − 12b22b3 − 2b5)e5

k

+ (28b52 + 34b2

2b4 + 37b2b23 + 5b6 − 73b3

2b3 − 17b3b4 − 13b2b5)e6k + O(e7

k))

.

(19)

Hence, from (15) and (19), we have

f (zk) f ′2(xk) = f ′3(η)(b2e2

k + 2(b22 + b3)e3

k + (7b2b3 + b32 + 3b4)e4

k + 2(5b2b4 + 2b22b3 + 3b2

3 + 2b5)e5k

+ (4b2b23 + 6b2

2b4 + b32b3 + 13b2b5 + 17b3b4 + 5b6)e6

k + O(e7k))

.(20)

9


Also, from (14), (15), and (16), one has

L(xk) = −2 f ′3(η)(ek + 4b2e2

k + 2(3b22 + 2b3)e3

k + (14b2b3 + 3b32 + 3b4)e4

k

+(14b2b4 + 11b22b3 + 9b2

3 + b5)e5k

+ 2(7b2b23 + 6b2

2b4 + 6b2b5 + 10b3b4 + b6)e6k + O(e7

k))

.

(21)

Therefore, combining (20) and (21), one can have

2 f (zk) f ′2(xk)L(xk)

= −b2ek − 2(b3 − b22)e

2k − (3b3

2 + 3b4 − 5b2b3)e3k

−(6b22b3 + 4b5 − 5b2b4 − 3b4

2 − 2b23)e

4k + O(e5

k).(22)

Furthermore, from (17) and (22), we get

xk − yk1 + 2 f (zk) f ′2(xk)L−1(xk)

= ek − (b32 − 2b2b3)e4

k − (12b22b3 − 5b2b4 − 4b4

2 − 4b23)e

5k + O(e6

k). (23)

So, substituting (23) into (10) in Method 4, one obtains

xk+1 = η + (b32 − 2b2b3)e4

k + O(e5k). (24)

Noticing that the (k + 1)-st error ek+1 = xk+1 − η, from (24) we have the following error equation

ek+1 = (b32 − 2b2b3)e4

k + O(e5k), (25)

which shows that Method 4 defined by (10) is at least fourth-order convergent according to Definition 4.We have shown Theorem 2.

Theorem 3. Suppose that f (x) is a function whose n-th derivative f (n)(x), n = 1, 2, 3, . . ., exists ina neighborhood of its root η with f ′(η) �= 0. If the initial approximation x0 is sufficiently close to η, then theMethod 5 defined by (12) is also at least fourth-order convergent with the following error equation

ek+1 = (b32 − b2b3)e4

k + O(e5k),

where ek = xk − η, k = 1, 2, 3, . . ., and the constants bn = anf ′(η) , an = f (n)(η)

n! , n = 1, 2, 3, . . ..

Proof of Theorem 3. Referring to (14) and (19) in the proof of Theorem 2, then, dividing f (zk) byf (xk)− f (zk) we see that

f (zk)f (xk)− f (zk)

= b2ek − 2(b22 − b3)e2

k − 3(2b2b3 − b32 − b4)e3

k − (3b42 + 4b2

3 + 8b2b4 − 11b22b3 − 4b5)e4

k

−(10b32b3 + 10b3b4 + 10b2b5 − 11b2b2

3 − 14b22b4 − 5b6)e5

k

−(221b42b3 + 16b3

3 + 78b2b3b4 + 27b22b5 − 73b6

2

−158b22b2

3 − 91b32b4 − 6b2

4 − 10b3b5 − 4b2b6)e6k + O(e7

k).

(26)

From (26), we obtain

f (xk)− f (zk)f (xk)−2 f (zk)

= 11− f (zk)

f (xk)− f (zk)

= 1 + b2ek + (2b3 − b22)e

2k + (3b4 − 2b2b3)e3

k

+(2b42 + 4b5 − 3b2

2b3 − 2b2b4)e4k + (14b3

2b3 + 2b3b4 + 5b6 − 5b52

−9b2b23 − 5b2

2b4 − 2b2b5)e5k + (77b6

2 + 192b22b2

3 + 121b32b4 + 15b2

4

+26b3b5 + 14b2b6 − 240b42b3 − 24b3

3 − 130b2b3b4 − 51b22b5)e6

k + O(e7k).

(27)

10


Multiplying (27) by (17) yields that

f (xk)− f (yk)

f (xk)− 2 f (yk)(xk − zk) = ek − (b3

2 − b2b3)e4k + 2(2b4

2 − 4b22b3 + b2

3 + b2b4)e5k + O(e6

k). (28)

Consequently, from (12) and the above Equation (28), and noticing the error ek+1 = xk+1 − η,we get the error equation as below:

ek+1 = (b32 − b2b3)e4

k + O(e5k). (29)

Thus, according to Definition 4, we have shown that Method 5 defined by (12) has fourth-orderconvergence at least. This completes the proof of Theorem 3.

Remark 6. Per iteration, Method 5 requires two evaluations of the function and one of its first-order derivative.If we consider the definition of efficiency index [3] as τ

√λ, where λ is the order of convergence of the method

and τ is the total number of new function evaluations (i.e., the values of f and its derivatives) per iteration,then Method 5 has the efficiency index equal to 3

√4 ≈ 1.5874, which is better than the ones of Halley iterative

method 3√

3 ≈ 1.4423 and Newton iterative method√

2 ≈ 1.4142.

5. Numerical Results

In this section, we present the results of numerical calculations to compare the efficiency ofthe proposed iterative methods (Methods 3–5) with Newton iterative method (Method 1, NIM forshort), Halley iterative method (Method 2, HIM for short) and a few classical variants defined inliteratures [19–22], such as the next iterative schemes with fourth-order convergence:

(i) Kou iterative method (KIM for short) [19].

xk+1 = xk − 2

1 +√

1− 2L f (xk)

f (xk)

f ′(xk), k = 0, 1, 2, . . . ,

where L f (xk) is defined by the equation as follows:

L f (xk) =f ′′ (xk − f (xk)/ (3 f ′(xk))) f (xk)

f ′2(xk).

(ii) Double-Newton iterative method (DNIM for short) [20].⎧⎨⎩ zk = xk − f (xk)f ′(xk)

,

xk+1 = xk − f (xk)f ′(xk)

− f (zk)f ′(zk)

, k = 0, 1, 2, . . . .

(iii) Chun iterative method (CIM for short) [21].⎧⎨⎩ zk = xk − f (xk)f ′(xk)

,

xk+1 = xk − f (xk)f ′(xk)

−(

1 + 2 f (zk)f (xk)

+ f 2(zk)f 2(xk)

)f (zk)f ′(xk)

, k = 0, 1, 2, . . . .

(iv) Jarratt-type iterative method (JIM for short) [22].⎧⎪⎨⎪⎩zk = xk − 2

3f (xk)f ′(xk)

,

xk+1 = xk − 4 f (xk)f ′(xk)+3 f ′(zk)

(1 + 9

16

(f ′(zk)f ′(xk)

− 1)2)

, k = 0, 1, 2, . . . .

11


In iterative process, we use the following stopping criteria for computer programs:

|xk+1 − xk| < ε and | f (xk+1)| < ε,

where the fixed tolerance ε is taken as 10−14. When the stopping criteria are satisfied, xk+1 can beregarded as the exact root η of the equation. Numerical experiments are performed in Mathematica10 environment with 64 digit floating point arithmetics (Digits: =64). Different test equations fi = 0,i = 1, 2, . . . , 5, the initial guess value x0, the number of iterations k + 1, the approximate root xk+1,the values of |xk+1 − xk| and | f (xk+1)| are given in Table 1. The following test equations are used inthe numerical results:

f1(x) = x3 − 11 = 0,

f2(x) = cos x− x = 0,

f3(x) = x3 + 4x2 − 25 = 0,

f4(x) = x2 − ex − 3x + 2 = 0,

f5(x) = (x + 2)ex − 1 = 0.

Table 1. Numerical results and comparison of various iterative methods.

Methods Equation x0 k + 1 xk+1 |xk+1 − xk| | f (xk+1)|NIM f1 = 0 1.5 7 2.22398009056931552116536337672215719652 1.1× 10−25 4.1× 10−47

HIM f1 = 0 1.5 5 2.22398009056931552116536337672215719652 1.7× 10−41 1.0× 10−46

Method 3 f1 = 0 1.5 4 2.22398009056931552116536337672215719652 8.3× 10−40 1.6× 10−48

Method 4 f1 = 0 1.5 4 2.22398009056931552116536337672215719652 8.3× 10−22 1.9× 10−47

Method 5 f1 = 0 1.5 4 2.22398009056931552116536337672215719652 7.5× 10−30 7.4× 10−45

KIM f1 = 0 1.5 4 2.22398009056931552116536337672215719652 8.5× 10−38 3.9× 10−48

DNIM f1 = 0 1.5 4 2.22398009056931552116536337672215719652 1.1× 10−25 1.1× 10−47

CIM f1 = 0 1.5 5 2.22398009056931552116536337672215719652 1.5× 10−41 6.6× 10−45

JIM f1 = 0 1.5 5 2.22398009056931552116536337672215719652 1.2× 10−45 4.3× 10−47

NIM f2 = 0 1 5 0.73908513321516064165531208767387340401 6.4× 10−21 1.5× 10−41

HIM f2 = 0 1 4 0.73908513321516064165531208767387340401 3.4× 10−29 5.1× 10−49

Method 3 f2 = 0 1 3 0.73908513321516064165531208767387340401 8.2× 10−19 7.5× 10−49

Method 4 f2 = 0 1 3 0.73908513321516064165531208767387340401 1.4× 10−17 9.4× 10−48

Method 5 f2 = 0 1 3 0.73908513321516064165531208767387340401 1.1× 10−18 7.5× 10−47

KIM f2 = 0 1 3 0.73908513321516064165531208767387340401 1.5× 10−20 8.3× 10−49

DNIM f2 = 0 1 3 0.73908513321516064165531208767387340401 6.4× 10−21 9.5× 10−48

CIM f2 = 0 1 3 0.73908513321516064165531208767387340401 2.2× 10−17 9.4× 10−48

JIM f2 = 0 1 3 0.73908513321516064165531208767387340401 7.4× 10−18 8.3× 10−49

NIM f3 = 0 3.5 7 2.03526848118195915354755041547361249916 6.4× 10−28 2.9× 10−47

HIM f3 = 0 3.5 5 2.03526848118195915354755041547361249916 2.0× 10−39 5.8× 10−47

Method 3 f3 = 0 3.5 4 2.03526848118195915354755041547361249916 2.0× 10−33 6.0× 10−47

Method 4 f3 = 0 3.5 4 2.03526848118195915354755041547361249916 2.0× 10−33 8.0× 10−46

Method 5 f3 = 0 3.5 4 2.03526848118195915354755041547361249916 3.4× 10−30 3.1× 10−45

KIM f3 = 0 3.5 4 2.03526848118195915354755041547361249916 4.3× 10−33 8.6× 10−47

DNIM f3 = 0 3.5 4 2.03526848118195915354755041547361249916 6.4× 10−28 9.9× 10−46

CIM f3 = 0 3.5 4 2.03526848118195915354755041547361249916 1.1× 10−20 9.6× 10−46

JIM f3 = 0 3.5 4 2.03526848118195915354755041547361249916 1.9× 10−22 1.1× 10−49

12


Table 1. Cont.

Methods Equation x0 k + 1 xk+1 |xk+1 − xk| | f (xk+1)|NIM f4 = 0 3.6 8 0.25753028543986076045536730493724178138 6.5× 10−29 3.5× 10−46

HIM f4 = 0 3.6 6 0.25753028543986076045536730493724178138 4.8× 10−37 1.6× 10−46

Method 3 f4 = 0 3.6 4 0.25753028543986076045536730493724178138 9.6× 10−14 2.7× 10−46

Method 4 f4 = 0 3.6 5 0.25753028543986076045536730493724178138 1.1× 10−36 3.3× 10−44

Method 5 f4 = 0 3.6 4 0.25753028543986076045536730493724178138 2.5× 10−19 4.9× 10−44

KIM f4 = 0 3.6 5 0.25753028543986076045536730493724178138 2.1× 10−14 8.9× 10−44

DNIM f4 = 0 3.6 4 0.25753028543986076045536730493724178138 2.6× 10−14 3.5× 10−46

CIM f4 = 0 3.6 4 0.25753028543986076045536730493724178138 2.8× 10−12 2.8× 10−46

JIM f4 = 0 3.6 5 0.25753028543986076045536730493724178138 9.7× 10−38 9.7× 10−46

NIM f5 = 0 3.5 11 −0.44285440100238858314132799999933681972 8.2× 10−22 7.7× 10−43

HIM f5 = 0 3.5 7 −0.44285440100238858314132799999933681972 2.2× 10−37 6.1× 10−45

Method 3 f5 = 0 3.5 5 −0.44285440100238858314132799999933681972 1.8× 10−24 3.4× 10−45

Method 4 f5 = 0 3.5 5 −0.44285440100238858314132799999933681972 5.3× 10−37 7.9× 10−44

Method 5 f5 = 0 3.5 6 −0.44285440100238858314132799999933681972 2.0× 10−42 3.9× 10−42

KIM f5 = 0 3.5 7 −0.44285440100238858314132799999933681972 3.6× 10−23 2.7× 10−42

DNIM f5 = 0 3.5 6 −0.44285440100238858314132799999933681972 8.2× 10−22 4.9× 10−45

CIM f5 = 0 3.5 7 −0.44285440100238858314132799999933681972 3.3× 10−37 8.6× 10−44

JIM f5 = 0 3.5 6 −0.44285440100238858314132799999933681972 9.3× 10−13 6.6× 10−46

6. Conclusions

In Section 3 of the paper, it is evident that we have obtained a few single-step iterative methodsincluding classical Newton’s method and Halley’s method, based on [1, n]-order Padé approximationof a function for finding a simple root of nonlinear equations. In order to avoid calculating the higherderivatives of the function, we have tried to improve the proposed iterative method by applyingapproximants of the second derivative and the third derivative. Hence, we have gotten a few modifiedtwo-step iterative methods free from the higher derivatives of the function. In Section 4, we havegiven theoretical proofs of the several methods. It is seen that any modified iterative method reachesthe convergence order 4. However, it is worth mentioning that Method 5 is free from second orderderivative and its efficiency index is 1.5874. Furthermore, in Section 5, numerical examples areemployed to illustrate the practicability of the suggested variants for finding the approximate roots ofsome nonlinear scalar equations. The computational results presented in Table 1 show that in almostall of the cases the presented variants converge more rapidly than Newton iterative method and Halleyiterative method, so that they can compete with Newton iterative method and Halley iterative method.Finally, for more nonlinear equations we tested, the presented variants have at least equal performancecompared to the other existing iterative methods that are of the same order.

Author Contributions: The contributions of all of the authors have been similar. All of them have worked togetherto develop the present manuscript.

Funding: This research was funded by the Natural Science Key Foundation of Education Department of AnhuiProvince (Grant No. KJ2013A183), the Project of Leading Talent Introduction and Cultivation in Colleges andUniversities of Education Department of Anhui Province (Grant No. gxfxZD2016270) and the Incubation Projectof National Scientific Research Foundation of Bengbu University (Grant No. 2018GJPY04).

Acknowledgments: The authors are thankful to the anonymous reviewers for their valuable comments.

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Alfio, Q.; Riccardo, S.; Fausto, S. Rootfinding for nonlinear equations. In Numerical Mathematics; Springer:New York, NY, USA, 2000; pp. 251–285, ISBN 0-387-98959-5.

2. Burden, A.M.; Faires, J.D.; Burden, R.L. Solutions of equations in one variable. In Numerical Analysis, 10th ed.;Cengage Learning: Boston, MA, USA, 2014; pp. 48–101, ISBN 978-1-305-25366-7.

13


3. Gautschi, W. Nonlinear equations. In Numerical Analysis, 2nd ed.; Birkhäuser: Boston, MA, USA, 2011;pp. 253–323, ISBN 978-0-817-68258-3.

4. Weerakoon, S.; Fernando, T.G.I. A variant of Newton’s method with accelerated third-order convergence.Appl. Math. Lett. 2000, 13, 87–93. [CrossRef]

5. Cordero, A.; Torregrosa, I.R. Variants of Newton’s method using fifth-order quadrature formulas.Appl. Math. Comput. 2007, 190, 686–698. [CrossRef]

6. Chun, C. Iterative methods improving Newton’s method by the decomposition method. Comput. Math. Appl.2005, 50, 1559–1568. [CrossRef]

7. Li, S.; Tan, J.; Xie, J.; Dong, Y. A new fourth-order convergent iterative method based on Thiele’s continuedfraction for solving equations. J. Inform. Comput. Sci. 2011, 8, 139–145.

8. Li, S.; Wang, R.; Zhang, X.; Xie, J.; Dong, Y. Halley’s iterative formula based on Padé approximation and itsmodifications. J. Inform. Comput. Sci. 2012, 9, 997–1004.

9. Abbasbandy, S.; Tan, Y.; Liao, S.J. Newton-homotopy analysis method for nonlinear equations.Appl. Math. Comput. 2007, 188, 1794–1800. [CrossRef]

10. Noor, M.A.; Khan, W.A. New iterative methods for solving nonlinear equation by using homotopyperturbation method. Appl. Math. Comput. 2012, 219, 3565–3574. [CrossRef]

11. Wang, X.; Qin, Y.; Qian, W.; Zhang, S.; Fan, X. A family of Newton type iterative methods for solvingnonlinear equations. Algorithms 2015, 8, 786–798. [CrossRef]

12. Sederberg, T.W.; Nishita, T. Curve intersection using Bézier clipping. Comput.-Aided Des. 1990, 22, 538–549.[CrossRef]

13. Barton, M.; Jüttler, B. Computing roots of polynomials by quadratic clipping. Comput. Aided Geom. Des.2007, 24, 125–141. [CrossRef]

14. Morlando, F. A class of two-step Newton’s methods with accelerated third-order convergence.Gen. Math. Notes 2015, 29, 17–26.

15. Thukral, R. New modification of Newton method with third-order convergence for solving nonlinearequations of type f (0) = 0. Am. J. Comput. Appl. Math. 2016, 6, 14–18. [CrossRef]

16. Rafiq, A.; Rafiullah, M. Some multi-step iterative methods for solving nonlinear equations.Comput. Math. Appl. 2009, 58, 1589–1597. [CrossRef]

17. Ali, F.; Aslam, W.; Ali, K.; Anwar, M.A.; Nadeem, A. New family of iterative methods for solving nonlinearmodels. Disc. Dyn. Nat. Soc. 2018, 1–12. [CrossRef]

18. Qureshi, U.K. A new accelerated third-order two-step iterative method for solving nonlinear equations.Math. Theory Model. 2018, 8, 64–68.

19. Kou, J.S. Some variants of Cauchy’s method with accelerated fourth-order convergence. J. Comput. Appl. Math.2008, 213, 71–78. [CrossRef]

20. Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA,1977; pp. 1–49.

21. Chun, C. Some fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comput.2008, 195, 454–459. [CrossRef]

22. Sharifi, M.; Babajee, D.K.R.; Soleymani, F. Finding the solution of nonlinear equations by a class of optimalmethods. Comput. Math. Appl. 2012, 63, 764–774. [CrossRef]

23. Wuytack, L. Padé approximants and rational functions as tools for finding poles and zeros of analyticalfunctions measured experimentally. In Padé Approximation and its Applications; Springer: New York, NY, USA,1979; pp. 338–351, ISBN 0-387-09717-1.

24. Xu, X.Y.; Li, J.K.; Xu, G.L. Definitions of Padé approximation. In Introduction to Padé Approximation;Shanghai Science and Technology Press: Shanghai, China, 1990; pp. 1–8, ISBN 978-7-532-31887-2.

c© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

14

mathematics

Article

Improving the Computational Efficiency of a Variantof Steffensen’s Method for Nonlinear Equations

Fuad W. Khdhr 1, Rostam K. Saeed 1 and Fazlollah Soleymani 2,*

1 Department of Mathematics, College of Science, Salahaddin University, Erbil, Iraq;[email protected] (F.W.K.); [email protected] (R.K.S.)

2 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS),Zanjan 45137-66731, Iran

* Correspondence: [email protected]

Received: 21 January 2019; Accepted: 18 March 2019; Published: 26 March 2019

Abstract: Steffensen-type methods with memory were originally designed to solve nonlinearequations without the use of additional functional evaluations per computing step. In this paper,a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, usingan acceleration technique via interpolation polynomials of appropriate degrees, the computationalefficiency index of this scheme is improved. It is discussed that the new scheme is quite fast andhas a high efficiency index. Finally, numerical investigations are brought forward to uphold thetheoretical discussions.

Keywords: iterative methods; Steffensen’s method; R-order; with memory; computational efficiency

1. Introduction

One of the commonly encountered topics in computational mathematics is to tackle solvinga nonlinear algebraic equation. The equation can be presented as in the scalar case f (x) = 0, or morecomplicated as a system of nonlinear algebraic equations. The procedure of finding the solutions (if itexists) cannot be done analytically. In some cases, the analytic techniques only give the real resultwhile its complex zeros should be found and reported. As such, numerical techniques are a viablechoice for solving such nonlinear problems. Each of the existing computational procedures has theirown domain of validity with some pros and cons [1,2].

Two classes of methods with the use of derivatives and without the use of derivatives are known tobe useful depending on the application dealing with [3]. In the derivative-involved methods, a largerattraction basin along with a simple coding effort for higher dimensional problems is at hand which,in derivative-free methods, the area of choosing the initial approximations is smaller and extending tohigher dimensional problems is via the application of a divided difference operator matrix, which isbasically a dense matrix. However, the ease in not computing the derivative and, subsequently,the Jacobians, make the application of derivative-free methods more practical in several problems [4–7].

Here, an attempt is made at developing a computational method which is not only efficient interms of the computational efficiency index, but also in terms of larger domains for the choice of theinitial guesses/approximations for starting the proposed numerical method.

The Steffensen’s method [8] for solving nonlinear scalar equations has quadratic convergence forsimple zeros and given by: ⎧⎪⎪⎪⎨⎪⎪⎪⎩

xk+1 = xk − f (xk)

f[xk, wk]

,

wk = xk + β f (xk), β ∈ R\{0}, k ≥ 0,

(1)



where the two-point divided difference is defined by:

f [xk, wk] =f (xk)− f (wx)

xk − wk,

This scheme needs two function evaluations per cycle. Scheme (1) shows an excellent tool forconstructing efficient iterative methods for nonlinear equations. This is because it is derivative-freewith a free parameter. This parameter can, first of all, enlarge the attraction basins of Equation (1)or any of its subsequent methods and, second, can directly affect the improvement of the R-order ofconvergence and the efficiency index.

Recalling that Kung and Traub conjectured that the iterative method without memory based on mfunctions evaluation per iteration attain the optimal convergence of order 2m−1 [9,10].

The term “with memory” means that the values of the function associated with the computedapproximations of the roots are used in subsequent iterations. This is unlike the term “without memory”in which the method only uses the current values to find the next estimate. As such, in a method withmemory, the calculated results up to the desired numbers of iterations should be stored and then calledto proceed.

Before proceeding the given idea to improve the speed of convergence, efficiency index, and theattraction basins, we provide a short literature by reviewing some of the existing methods withaccelerated convergence order. Traub [11] proposed the following two-point method with memory oforder 2.414: ⎧⎪⎪⎨⎪⎪⎩

xk+1 = xk − f (xk)

f [xk, xk + βk f (xk)],

βk =−1

f [xk, zk−1],

(2)

where zk−1 = xk−1 + βk−1 f (xk−1), and β0 = −sign( f ′(x0)) or − 1f [x0, x0 + f (x0)]

. This is one of the

pioneering and fundamental methods with memory for solving nonlinear equations.

Džunic in [12] suggested an effective bi-parametric iterative method with memory of12

(3 +

√17)

R-order of convergence as follows:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩wk = xk + βk f (xk),

βk = − 1N′2(xk)

, ζk = −N′′

3 (wk)

2N′3(wk), k ≥ 1,

xk+1 = xk − f (xk)

f [xk, wk] + ζk f (wk)k ≥ 0.

(3)

Moreover, Džunic and Petkovic [13] derived the following cubically convergent Steffensen-likemethod with memory: ⎧⎪⎪⎨⎪⎪⎩

xk+1 = xk − f (xk)

f [xk, xk + βk f (xk)],

βk =−1

f [xk, zk−1]+ f [xk, xk−1]+ f [xk−1, zk−1],

(4)

where zk−1 = xk−1 + βk−1 f (xk−1) depending on the second-order Newton interpolation polynomial.Various Steffensen-type methods are proposed in [14–17].In fact, it is possible to improve the performance of the aforementioned method by considering

several more sub-steps and improve the computational efficiency index via multi-step iterativemethods. However, this procedure is more computational burdensome. Thus, the motivation hereis to know that is it possible to improve the performance of numerical methods in terms of thecomputational efficiency index, basins of attraction, and the rate of convergence without adding moresub-steps and propose a numerical method as a one-step solver.

16


Hence, the aim of this paper is to design a one-step method with memory which is quitefast and has an improved efficiency index, based on the modification of the one-step method ofSteffensen (Equation (1)) and increase the convergence order to 3.90057 without any additionalfunctional evaluations.

The rest of this paper is ordered as follows: In Section 2, we develop the one-point Steffensen-typeiterative scheme (Equation (1)) with memory which was proposed by [18]. We present the maingoal in Section 3 by approximating the acceleration parameters involved in our contributed schemeby Newton’s interpolating polynomial and, thus, improve the convergence R-order. The numericalreports are suggested in Section 4 to confirm the theoretical results. Some discussions are given inSection 5.

2. An Iterative Method

The following iterative method without memory was proposed by [18]:⎧⎨⎩ wk = xk − β f (xk),

xk+1 = xk − f (xk)

f [xk, wk]

(1 + ζ

f (wk)

f [xk, wk]

), ζ ∈ R,

(5)

with the following error equation to improve the performance of (1) in terms of having morefree parameters:

ek+1 = −(−1 + β f ′(α))(c2 − ζ)e2

k + O(

e3k

), (6)

where ci =1i!

f (i)(α)f ′(α) . Using the error Equation (6), to derive Steffensen-type iterative methods with

memory, we calculate the following parameters: β = βk, ζ = ζk, by the formula:⎧⎨⎩ βk =1

f ′(xα),

ζk = c2,(7)

for k = 1, 2, 3, · · · , while f ′(xα), c2 are approximations to f (α) and c2, respectively; where α is a simplezero of f (x). In fact, Equation (7) shows a way to minimize the asymptotic error constant of Equation (6)by making this coefficient closer and closer to zero when the iterative method is converging to thetrue solution.

The initial estimates β0 and ζ0 must be chosen before starting the process of iterations. We statethe Newton’s interpolating polynomial of fourth and fifth-degree passing through the saved pointsas follows: {

N4(t) = N4(t; xk, wk−1, xk−1, wk−2, xk−2),N5(t) = N5(t; wk, xk, wk−1, xk−1, wk−2, xk−2).

(8)

Recalling that N(t) is an interpolation polynomial for a given set of data points also known as theNewton’s divided differences interpolation polynomial because the coefficients of the polynomial arecalculated using Newton’s divided differences method. For instance, here the set of data points forN4(t) are {{xk, f (xk)}, {wk−1, f (wk−1)}, {xk−1, f (xk−1)}, {wk−2, f (wk−2)} , {xk−2, f (xk−2)}}.

Now, using some modification on Equation (5) we present the following scheme:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩wk = xk − βk f (xk),

βk =1

N′4(xk), ζk =

N′′5 (wk)

2N′5(wk), k ≥ 2,

xk+1 = xk − f (xk)

f [xk, wk]

(1 + ζk

f (wk)

f [xk, wk]

), k ≥ 0.

(9)

17


Noting that the accelerator parameters βk, ζk are getting updated and then used in the iterativemethod right after the second iterations, viz, k ≥ 2. This means that the third line of Equation (9) isimposed at the beginning and after that the computed values are stored and used in the subsequentiterates. For k = 1, the degree of Newton interpolation polynomials would be two and three. However,for k ≥ 2, interpolations of degrees four and five as given in Equation (8) can be used to increase theconvergence order.

Additionally speaking, this acceleration of convergence would be attained without the use anymore functional evaluations as well as imposing more steps. Thus, the proposed scheme with memory(Equation (9)) can be attractive for solving nonlinear equations.

3. Convergence Analysis

In this section, we show the convergence criteria of Equation (9) using Taylor’s series expansionand several extensive symbolic computations.

Theorem 1. Let the function f (x) be sufficiently differentiable in a neighborhood of its simple zero α. If an initialapproximation x0 is necessarily close to α. Then, R-order of convergence for the one-step method (Equation (9))with memory is 3.90057.

Proof. The proof is done using the definition of the error equation as the difference between thek-estimate and the exact zero along with symbolic computations. Let the sequence {xk} and {wk}have convergence orders r and p, respectively. Namely,

ek+1 ∼ erk, (10)

and:ew,k ∼ ep

k , (11)

Therefore, using Equations (10) and (11), we have:

ek+1 ∼ erk ∼ er2

k−1 ∼ er3

k−2, (12)

and:ew,k ∼ ep

k ∼ (erk−1)

p ∼ epr2

k−2. (13)

The associated error equations to the accelerating parameters βk and ζk for Equation (9) can nowbe written as follows:

ew,k ∼ (−1 + βk f ′(α))ek , (14)

and:ek+1 ∼ −(−1 + βk f ′(α))(c2 − ζk)e2

k . (15)

On the other hand, by using a symbolic language and extensive computations one can find thefollowing error terms for the involved terms existing in the fundamental error Equation (6):

−1 + βk f ′(α) ∼ c5ek−2ek−1ew,k−1ew,k−2, (16)

c2 − ζk ∼ c6ek−2ek−1ew,k−1ew,k−2 (17)

Combining Equations (14)–(17), we get that:

ew,k ∼ er2+pr+r+p+1k−2 , (18)

ek+1 ∼ e2(r2+pr+r+p+1)k−2 . (19)

18


We now compare the left and right hand side of Equations (12)–(19) and Equations (13)–(18),respectively. Thus, we have the following nonlinear system of equations in order to find the finalR-orders: {

r2 p− (r2 + pr + r + p + 1)= 0,

r3 − 2(r2 + pr + r + p + 1

)= 0.

(20)

The positive real solution of (20) is r = 3.90057 and p = 1.9502. Therefore, the convergenceR-order for Equation (9) is 3.90057. �

Since improving the convergence R-order is useless if the whole computational method isexpensive, basically researcher judge on a new scheme based upon its computational efficiencyindex which is a tool in order to provide a trade-off between the whole computational cost and theattained R-order. Assuming the cost of calculating each functional evaluation is one, we can use thedefinition of efficiency index as EI = p1/θ , θ is the whole computational cost [19].

The computational efficiency index of Equation (9) is 3.9005712 ≈ 1.97499 ≈ 2, which is clearly

higher than efficiency index 212 ≈ 1.4142 of Newton’s and Steffensen’s methods, 3.56155

12 ≈ 1.8872 of

(3) 31/2 ≈ 1.73205 of Equation (4).However, this improved computational efficiency is reported by ignoring the number of

multiplication and division per computing cycle. By imposing a slight weight for such calculationsone may once again obtain the improved computational efficiency of (9) in contrast to the existingschemes of the same type.

4. Numerical Computations

In this section, we compare the convergence performance of Equation (9), with three well-knowniterative methods for solving four test problems numerically carried out in Mathematica 11.1. [20].

We denote Equations (1), (3), (5) and (9) with SM, DZ, PM, M4, respectively. We compare theour method with different methods, using β0 = 0.1 and ζ0 = 0.1. Here, the computational order ofconvergence (coc) has been computed by the following formula [21]:

coc =ln|( f (xk)/ f (xk−1)|

ln|( f (xk−1)/ f (xk−2)| (21)

Recalling that using a complex initial approximation, one is able to find the complex roots of thenonlinear equations using (9).

Experiment 1. Let us consider the following nonlinear test function:

f1(x) = (x− 2 tan(x))(

x3 − 8), (22)

where α = 2 and x0 = 1.7.

Experiment 2. We take into account the following nonlinear test function:

f2(x) = (x− 1)(

x10 + x3 + 1)

sin(x), (23)

where α = 1 and x0 = 0.7.

Experiment 3. We consider the following test problem now:

f3(x) = −x3

2 + 2 tan−1(x) + 1, (24)

where α ≈ 1.8467200 and x0 = 4.

19


Experiment 4. The last test problem is taken into consideration as follows:

f4(x) = tan−1(exp(x + 2) + 1) + tanh(exp(−x cos(x)))− sin(πx), (25)

where α ≈ −3.6323572··· and x0 = −4.1.

Tables 1–4 show that the proposed Equation (9) is of order 3.90057 and it is obviously believed tobe of more advantageous than the other methods listed due to its fast speed and better accuracy.

For better comparisons, we present absolute residual errors | f (x)|, for each test function whichare displayed in Tables 1–4. Additionally, we compute the computational order of convergence.Noting that we have used multiple precision arithmetic considering 2000 significant digits to observeand the asymptotic error constant and the coc as obviously as possible.

The results obtained by our proposed Equation (M4) are efficient and show better performancethan other existing methods.

A significant challenge of executing high-order nonlinear solvers is in finding initial approximationto start the iterations when high accuracy calculating is needed.

Table 1. Result of comparisons for the function f1.

Methods |f1(x3)| |f1(x4)| |f1(x5)| |f1(x6)| coc

SM 4.1583 3.0743 1.4436 0.25430 2.00DZ 0.13132 2.0026× 10−7 1.0181× 10−27 7.1731× 10−99 3.57PM 1.8921× 10−6 4.5864× 10−24 1.0569× 10−88 7.5269× 10−318 3.55M4 9.1741× 10−6 3.3242× 10−26 4.4181× 10−103 1.1147× 10−404 3.92



SM − − − − −DZ 0.14774 0.0016019. 1.3204× 10−10 1.5335× 10−35 3.56PM 2.1191× 10−10 8.0792× 10−35 1.9037× 10−121 3.7062× 10−430 3.56M4 5.9738× 10−15 4.1615× 10−57 1.7309× 10−220 1.8231× 10−857 3.90



SM 0.042162 0.00012627 1.1589× 10−9 9.7638× 10−20 2.00DZ 1.0219× 10−11 4.4086× 10−44 1.6412× 10−157 1.5347× 10−562 3.57PM 7.9792× 10−8 3.712× 10−30 4.9556× 10−108 2.9954× 10−386 3.57M4 4.4718× 10−6 2.9187× 10−25 4.7057× 10−101 1.0495× 10−395 3.89

To discuss further, mostly based on interval mathematics, one can find a close enough guess tostart the process. There are some other ways to determine the real initial approximation to start theprocess. An idea of finding such initial guesses given in [22] is based on the useful commands inMathematica 11.1 NDSolve [] for the nonlinear function on the interval D = [a, b].

Following this the following piece of Mathematica code could give a list of initial approximationsin the working interval for Experiment 4:

20


ClearAll[“Global`*”] (*Defining the nonlinear function.*) f[x_]:=ArcTan[Exp[x+2]+1]+Tanh[Exp[ x Cos[x]]] Sin[Pi x]; (*Defining the interval.*) a= 4.; b=4.; (*Find the list of initial estimates.*) Zeros = Quiet@Reap[soln=y[x]/.First[NDSolve[{y’[x] ==Evaluate[D[f[x],x]],y[b]==(f[b])},y[x],{x,a,b}, Method->{“EventLocator”,”Event”->y[x], “EventAction”:>Sow[{x,y[x]}]}]]][[2,1]]; initialPoints = Sort[Flatten[Take[zeros,Length[zeros],1]]]

To check the position of the zero and the graph of the function, we can use the following code toobtain Figure 1.

Length[initialPoints] Plot[f[x],{x,a,b}, Epilog->{PointSize[Medium], Red, Point[zeros]},PlotRange->All, PerformanceGoal->“Quality”, PlotStyle->{Thick, Blue}]



SM 0.00001166 3.7123× 10−10 3.7616× 10−19 3.8622× 10−37 2.00DZ 1.6× 10−13 6.9981× 10−47 1.0583× 10−164 7.0664× 10−585 3.57PM 3.0531× 10−11 3.2196× 10−38 3.7357× 10−134 6.5771× 10−476 3.56M4 2.5268× 10−13 1.5972× 10−49 2.8738× 10−191 1.6018× 10−744 3.90

Figure 1. The plot of the nonlinear function in Experiment 4 along with its roots colored in red.

As a harder test problem, for the nonlinear function g(x) = 2x + 0.5 sin(20π x) − x2, we cansimply find a list of estimates as initial guesses using the above piece of codes as follows: {−0.185014,−0.162392, −0.0935912, −0.0535277, 6.73675 × 10−9, 0.0533287, 0.0941576, 0.160021, 0.188066, 0.269075,0.279428, 1.76552, 1.78616, 1.8588, 1.89339,1.95294, 2., 2.04692, 2.10748, 2.13979, 2.2228, 2.22471}. The plotof the function in this case is brought forward in Figure 2.

We observe that the two self-accelerating parameters β0 and ζ0 have to be selected before theiterative procedure is started. That is, they are calculated by using information existing from thepresent and previous iterations (see, e.g., [23]). The initial estimates β0 and ζ0 should be preserved asprecise small positive values. We use β0 = ζ0 = 0.1 whenever required.

21


After a number of iterates, the (nonzero) free parameters start converging to a particular valuewhich makes the coefficient of Equation (6) zero as well as make the numerical scheme to convergewith high R-order.

Figure 2. The behavior of the function g and the position of its roots (the red dots show the location ofthe zeros of the nonlinear functions).

5. Ending Comments

In this paper, we have constructed a one-step method with memory to solve nonlinearequations. By using two self-accelerator parameters our scheme equipped with Newton’s interpolationpolynomial without any additional functional calculation possesses the high computational efficiencyindex 1.97499, which is higher than many of the existing methods.

The efficacy of our scheme is confirmed by some of numerical examples. The results in Tables 1–4shows that our method (Equation (M4)) is valuable to find an adequate estimate of the exact solutionof nonlinear equations.

Author Contributions: The authors contributed equally to this paper.

Funding: This research received no external funding.

Acknowledgments: The authors are thankful to two anonymous referees for careful reading and valuablecomments which improved the quality of this manuscript.


References

1. Cordero, A.; Hueso, J.L.; Martinez, E.; Torregrosa, J.R. Steffensen type methods for solving nonlinearequations. J. Comput. Appl. Math. 2012, 236, 3058–3064. [CrossRef]

2. Soleymani, F. Some optimal iterative methods and their with memory variants. J. Egypt. Math. Soc. 2013,21, 133–141. [CrossRef]

3. Praks, P.; Brkic, D. Choosing the optimal multi-point iterative method for the Colebrook Flow frictionequation. Processes 2018, 6, 130. [CrossRef]

4. Zafar, F.; Cordero, A.; Torregrosa, J.R. An efficient family of optimal eighth-order multiple root finders.Mathematics 2018, 6, 310. [CrossRef]

5. Saeed, R.K.; Aziz, K.M. An iterative method with quartic convergence for solving nonlinear equations.Appl. Math. Comput. 2008, 202, 435–440. [CrossRef]

6. Saeed, R.K. Six order iterative method for solving nonlinear equations. World Appl. Sci. J. 2010, 11, 1393–1397.7. Torkashvand, V.; Lotfi, T.; Araghi, M.A.F. A new family of adaptive methods with memory for solving

nonlinear equations. Math. Sci. 2019, 1–20.8. Noda, T. The Steffensen iteration method for systems of nonlinear equations. Proc. Jpn. Acad. 1987, 63, 186–189.9. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 1974,

21, 634–651. [CrossRef]

22


10. Ahmad, F. Comment on: On the Kung-Traub conjecture for iterative methods for solving quadratic equations.Algorithms 2016, 9, 30. [CrossRef]

11. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.12. Džunic, J. On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms 2013,

63, 549–569. [CrossRef]13. Džunic, J.; Petkovic, M.S. On generalized biparametric multipoint root finding methods with memory.

J. Comput. Appl. Math. 2014, 255, 362–375. [CrossRef]14. Zheng, O.; Wang, J.; Zhang, P.L. A Steffensen-like method and its higher-order variants. Appl. Math. Comput.

2009, 214, 10–16. [CrossRef]15. Lotfi, T.; Tavakoli, E. On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator

parameter. Sci. World J. 2014, 2014, 769758. [CrossRef]16. Zheng, O.P.; Zhao, L.; Ma, W. Variants of Steffensen-Secant method and applications. Appl. Math. Comput.

2010, 216, 3486–3496. [CrossRef]17. Petkovic, M.S.; Ilic, S.; Džunic, J. Derivative free two-point methods with and without memory for solving

nonlinear equations. Appl. Math. Comput. 2010, 217, 1887–1895.18. Khaksar Haghani, F. A modiffied Steffensen’s method with memory for nonlinear equations. Int. J. Math.

Model. Comput. 2015, 5, 41–48.19. Howk, C.L.; Hueso, J.L.; Martinez, E.; Teruel, C. A class of efficient high-order iterative methods with

memory for nonlinear equations and their dynamics. Math. Meth. Appl. Sci. 2018, 1–20. [CrossRef]20. Cliff, H.; Kelvin, M.; Michael, M. Hands-on Start to Wolfram Mathematica and Programming with the Wolfram

Language, 2nd ed.; Wolfram Media, Inc.: Champaign, IL, USA, 2016; ISBN 9781579550127.21. Weerakoon, S.; Fernando, T.G.I. A variant of Newton’s method with accelerated third-order convergence.

Appl. Math. Lett. 2000, 13, 87–93. [CrossRef]22. Soleymani, F.; Shateyi, S. Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal.

2012, 2012, 318165. [CrossRef]23. Zaka, M.U.; Kosari, S.; Soleymani, F.; Khaksar, F.H.; Al-Fhaid, A.S. A super-fast tri-parametric iterative

method with memory. Appl. Math. Comput. 2016, 289, 486–491.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

23

mathematics

Article

Optimal Fourth, Eighth and Sixteenth Order Methodsby Using Divided Difference Techniques and TheirBasins of Attraction and Its Application

Yanlin Tao 1,† and Kalyanasundaram Madhu 2,*,†

1 School of Computer Science and Engineering, Qujing Normal University, Qujing 655011, China;[email protected]

2 Department of Mathematics, Saveetha Engineering College, Chennai 602105, India* Correspondence: [email protected]; Tel.: +91-9840050042† These authors contributed equally to this work.

Received: 26 February 2019; Accepted: 26 March 2019; Published: 30 March 2019

Abstract: The principal objective of this work is to propose a fourth, eighth and sixteenth orderscheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourthorder method uses two evaluations of the function and one evaluation of the first derivative; theeighth order method uses three evaluations of the function and one evaluation of the first derivative;and sixteenth order method uses four evaluations of the function and one evaluation of the firstderivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition,the theoretical convergence properties of our schemes are fully explored with the help of the maintheorem that demonstrates the convergence order. The performance and effectiveness of our optimaliteration functions are compared with the existing competitors on some standard academic problems.The conjugacy maps of the presented method and other existing eighth order methods are discussed,and their basins of attraction are also given to demonstrate their dynamical behavior in the complexplane. We apply the new scheme to find the optimal launch angle in a projectile motion problem andPlanck’s radiation law problem as an application.

Keywords: non-linear equation; basins of attraction; optimal order; higher order method; computationalorder of convergence

MSC: 65H05, 65D05, 41A25

1. Introduction

One of the most frequent problems in engineering, scientific computing and applied mathematics,in general, is the problem of solving a nonlinear equation f (x) = 0. In most of the cases, wheneverreal problems are faced, such as weather forecasting, accurate positioning of satellite systems in thedesired orbit, measurement of earthquake magnitudes and other high-level engineering problems,only approximate solutions may get resolved. However, only in rare cases, it is possible to solve thegoverning equations exactly. The most familiar method of solving non linear equation is Newton’siteration method. The local order of convergence of Newton’s method is two and it is an optimalmethod with two function evaluations per iterative step.

In the past decade, several higher order iterative methods have been developed and analyzed forsolving nonlinear equations that improve classical methods such as Newton’s method, Chebyshevmethod, Halley’s iteration method, etc. As the order of convergence increases, so does the numberof function evaluations per step. Hence, a new index to determine the efficiency called the efficiencyindex is introduced in [1] to measure the balance between these quantities. Kung-Traub [2] conjectured



that the order of convergence of any multi-point without memory method with d function evaluationscannot exceed the bound 2d−1, the optimal order. Thus the optimal order for three evaluations periteration would be four, four evaluations per iteration would be eight, and so on. Recently, some fourthand eighth order optimal iterative methods have been developed (see [3–14] and references therein).A more extensive list of references as well as a survey on the progress made in the class of multi-pointmethods is found in the recent book by Petkovic et al. [11].

This paper is organized as follows. An optimal fourth, eighth and sixteenth order methods aredeveloped by using divided difference techniques in Section 2. In Section 3, convergence order isanalyzed. In Section 4, tested numerical examples to compare the proposed methods with otherknown optimal methods. The problem of Projectile motion is discussed in Section 5 where thepresented methods are applied on this problem with some existing ones. In Section 6, we obtain theconjugacy maps of these methods to make a comparison from dynamical point of view. In Section7, the proposed methods are studied in the complex plane using basins of attraction. Section 8 givesconcluding remarks.

2. Design of an Optimal Fourth, Eighth and Sixteenth Order Methods

Definition 1 ([15]). If the sequence {xn} tends to a limit x∗ in such a way that

limn→∞

xn+1 − x∗

(xn − x∗)p = C

for p ≥ 1, then the order of convergence of the sequence is said to be p, and C is known as the asymptotic errorconstant. If p = 1, p = 2 or p = 3, the convergence is said to be linear, quadratic or cubic, respectively.Let en = xn − x∗, then the relation

en+1 = C epn + O

(ep+1

n

)= O

(ep

n

). (1)

is called the error equation. The value of p is called the order of convergence of the method.

Definition 2 ([1]). The Efficiency Index is given by

EI = p1d , (2)

where d is the total number of new function evaluations (the values of f and its derivatives) per iteration.

Let xn+1 = ψ(xn) define an Iterative Function (IF). Let xn+1 be determined by new information atxn, φ1(xn), ..., φi(xn), i ≥ 1. No old information is reused. Thus,

xn+1 = ψ(xn, φ1(xn), ..., φi(xn)). (3)

Then ψ is called a multipoint IF without memory.The Newton (also called Newton-Raphson) IF (2ndNR) is given by

ψ2nd NR(x) = x− f (x)f ′(x)

. (4)

The 2ndNR IF is one-point IF with two function evaluations and it satisfies the Kung-Traubconjecture with d = 2. Further, EI2nd NR = 1.414.

25


2.1. An Optimal Fourth Order Method

We attempt to get a new optimal fourth order IF as follows, let us consider two step Newton’smethod

ψ4th NR(x) = ψ2nd NR(x)− f (ψ2nd NR(x))f ′(ψ2nd NR(x))

. (5)

The above one is having fourth order convergence with four function evaluations. But, this is notan optimal method. To get an optimal, need to reduce a function and preserve the same convergenceorder, and so we estimate f ′(ψ2nd NR(x)) by the following polynomial

q(t) = a0 + a1(t− x) + a2(t− x)2, (6)

which satisfiesq(x) = f (x), q′(x) = f ′(x), q(ψ2nd NR(x)) = f (ψ2nd NR(x)).

On implementing the above conditions on Equation (6), we obtain three unknowns a0, a1 and a2.Let us define the divided differences

f [y, x] =f (y)− f (x)

y− x, f [y, x, x] =

f [y, x]− f ′(x)y− x

.

From conditions, we get a0 = f (x), a1 = f ′(x) and a2 = f [ψ2nd NR(x), x, x], respectively, by usingdivided difference techniques. Now, we have the estimation

f ′(ψ2nd NR(x)) ≈ q′(ψ2nd NR(x)) = a1 + 2a2(ψ2th NR(x)− x).

Finally, we propose a new optimal fourth order method as

ψ4thYM(x) = ψ2nd NR(x)− f (ψ2nd NR(x))f ′(x) + 2 f [ψ2nd NR(x), x, x](ψ2th NR(x)− x)

. (7)

The efficiency of the method (7) is EI4thYM = 1.587.

2.2. An Optimal Eighth Order Method

Next, we attempt to get a new optimal eighth order IF as following way

ψ8thYM(x) = ψ4thYM(x)− f (ψ4thYM(x))f ′(ψ4thYM(x))

.

The above one is having eighth order convergence with five function evaluations. But, this is notan optimal method. To get an optimal, need to reduce a function and preserve the same convergenceorder, and so we estimate f ′(ψ4thYM(x)) by the following polynomial

q(t) = b0 + b1(t− x) + b2(t− x)2 + b3(t− x)3, (8)

which satisfies

q(x) = f (x), q′(x) = f ′(x), q(ψ2nd NR(x)) = f (ψ2nd NR(x)), q(ψ4thYM(x)) = f (ψ4thYM(x)).

On implementing the above conditions on (8), we obtain four linear equations with four unknownsb0, b1, b2 and b3. From conditions, we get b0 = f (x) and b1 = f ′(x). To find b2 and b3, we solve thefollowing equations:

f (ψ2nd NR(x)) = f (x) + f ′(x)(ψ2nd NR(x)− x) + b2(ψ2nd NR(x)− x)2 + b3(ψ2nd NR(x)− x)3

f (ψ4thYM(x)) = f (x) + f ′(x)(ψ4thYM(x)− x) + b2(ψ4thYM(x)− x)2 + b3(ψ4thYM(x)− x)3.

26


Thus by applying divided differences, the above equations simplifies to

b2 + b3(ψ2nd NR(x)− x) = f [ψ2nd NR(x), x, x] (9)

b2 + b3(ψ4thYM(x)− x) = f [ψ4thYM(x), x, x] (10)

Solving Equations (9) and (14), we have

b2 =f [ψ2nd NR(x), x, x](ψ4thPM(x)− x)− f [ψ4thYM(x), x, x](ψ2nd NR(x)− x)

ψ4thYM(x)− ψ2nd NR(x),

b3 =f [ψ4thYM(x), x, x]− f [ψ2nd NR(x), x, x]

ψ4thYM(x)− ψ2nd NR(x).

(11)

Further, using Equation (11), we have the estimation

f ′(ψ4thYM(x)) ≈ q′(ψ4thYM(x)) = b1 + 2b2(ψ4thYM(x)− x) + 3b3(ψ4thYM(x)− x)2.

Finally, we propose a new optimal eighth order method as

ψ8thYM(x) = ψ4thYM(x)− f (ψ4thYM(x))f ′(x) + 2b2(ψ4thYM(x)− x) + 3b3(ψ4thYM(x)− x)2 . (12)

The efficiency of the method (12) is EI8thYM = 1.682. Remark that the method is seems a particularcase of the method of Khan et al. [16], they used weight function to develop their methods. Whereaswe used finite difference scheme to develop proposed methods. We can say the methods 4thYM and8thYM are reconstructed of Khan et al. [16] methods.

2.3. An Optimal Sixteenth Order Method

Next, we attempt to get a new optimal sixteenth order IF as following way

ψ16thYM(x) = ψ8thYM(x)− f (ψ8thYM(x))f ′(ψ8thYM(x))

.

The above one is having eighth order convergence with five function evaluations. However,this is not an optimal method. To get an optimal, need to reduce a function and preserve the sameconvergence order, and so we estimate f ′(ψ8thYM(x)) by the following polynomial

q(t) = c0 + c1(t− x) + c2(t− x)2 + c3(t− x)3 + c4(t− x)4, (13)

which satisfies

q(x) = f (x), q′(x) = f ′(x), q(ψ2nd NR(x)) = f (ψ2nd NR(x)),

q(ψ4thYM(x)) = f (ψ4thYM(x)), q(ψ8thYM(x)) = f (ψ8thYM(x)).

On implementing the above conditions on (13), we obtain four linear equations with fourunknowns c0, c1, c2 and c3. From conditions, we get c0 = f (x) and c1 = f ′(x). To find c2, c3 and c4, wesolve the following equations:

f (ψ2nd NR(x)) = f (x) + f ′(x)(ψ2nd NR(x)− x) + c2(ψ2nd NR(x)− x)2 + c3(ψ2nd NR(x)− x)3 + c4(ψ2nd NR(x)− x)4

f (ψ4thYM(x)) = f (x) + f ′(x)(ψ4thYM(x)− x) + c2(ψ4thYM(x)− x)2 + c3(ψ4thYM(x)− x)3 + c4(ψ4thYM(x)− x)4

f (ψ8thYM(x)) = f (x) + f ′(x)(ψ8thYM(x)− x) + c2(ψ8thYM(x)− x)2 + c3(ψ8thYM(x)− x)3 + c4(ψ8thYM(x)− x)4.

27


Thus by applying divided differences, the above equations simplifies to

c2 + c3(ψ2nd NR(x)− x) + c4(ψ2nd NR(x)− x)2 = f [ψ2nd NR(x), x, x]

c2 + c3(ψ4thYM(x)− x) + c4(ψ4thYM(x)− x)2 = f [ψ4thYM(x), x, x] (14)

c2 + c3(ψ8thYM(x)− x) + c4(ψ8thYM(x)− x)2 = f [ψ8thYM(x), x, x]

Solving Equation (14), we have

c2 =

(f [ψ2nd NR(x), x, x]

(− S2

2S3 + S2S23

)+ f [ψ4thYM(x), x, x]

(S2

1S3 − S1S23


(− S2

1S2 + S1S22

))−S2

1S2 + S1S22 + S2

1S3 − S22S3 − S1S2

3 + S2S23

,

c3 =


(S2

2 − S23


(− S2

1 + S23


(S2

1 − S22

))−S2

1S2 + S1S22 + S2

1S3 − S22S3 − S1S2

3 + S2S23

,

c4 =


(− S2 + S3


(S1 − S3


(− S1 + S2

))−S2

1S2 + S1S22 + S2

1S3 − S22S3 − S1S2

3 + S2S23

,

S1 = ψ2nd NR(x)− x, S2 = ψ4thYM(x)− x, S3 = ψ8thYM(x)− x.

(15)

Further, using Equation (15), we have the estimation

f ′(ψ8thYM(x)) ≈ q′(ψ8thYM(x)) = c1 + 2c2(ψ8thYM(x)− x) + 3c3(ψ8thYM(x)− x)2 + 4c4(ψ8thYM(x)− x)3.

Finally, we propose a new optimal sixteenth order method as

ψ16thYM(x) = ψ8thYM(x)− f (ψ8thYM(x))f ′(x) + 2c2(ψ8thYM(x)− x) + 3c3(ψ8thYM(x)− x)2 + 4c4(ψ8thYM(x)− x)3 . (16)

The efficiency of the method (16) is EI16thYM = 1.741.


In this section, we prove the convergence analysis of proposed IFs with help of Mathematicasoftware.

Theorem 1. Let f : D ⊂ R→ R be a sufficiently smooth function having continuous derivatives. If f (x) hasa simple root x∗ in the open interval D and x0 chosen in sufficiently small neighborhood of x∗, then the method4thYM IFs (7) is of local fourth order convergence, and the 8thYM IFs (12) is of local eighth order convergence.

Proof. Let e = x− x∗ and c[j] =f (j)(x∗)j! f ′(x∗) , j = 2, 3, 4, .... Expanding f (x) and f ′(x) about x∗ by Taylor’s

method, we have

f (x) = f ′(x∗)(

e + e2c[2] + e3c[3] + e4c[4] + e5c[5] + e6c[6] + e7c[7] + e8c[8] + . . .)

(17)

and

f ′(x) = f ′(x∗)(

1 + 2e c[2] + 3e2c[3] + 4e3c[4] + 5e4c[5] + 6e5c[6] + 7e6c[7] + 8e7c[8] + 9e8c[9] + . . .)

(18)

28


Thus,

ψ2nd NR(x) = x∗ + c[2]e2 +(− 2c[2]2 + 2c[3]

)e3 +

(4c[2]3 − 7c[2]c[3] + 3c[4]

)e4 +

(− 8c[2]4

+ 20c[2]2c[3]− 6c[3]2 − 10c[2]c[4] + 4c[5])

e5 +(

16c[2]5 − 52c[2]3c[3] + 28c[2]2c[4]− 17c[3]c[4]

+ c[2](33c[3]2 − 13c[5]) + 5c[6])

e6 − 2(

16c[2]6 − 64c[2]4c[3]− 9c[3]3 + 36c[2]3c[4] + 6c[4]2 + 9c[2]2(7c[3]2

− 2c[5]) + 11c[3]c[5] + c[2](−46c[3]c[4] + 8c[6])− 3c[7])

e7 +(

64c[2]7 − 304c[2]5c[3]

+ 176c[2]4c[4] + 75c[3]2c[4] + c[2]3(408c[3]2 − 92c[5])− 31c[4]c[5]− 27c[3]c[6]

+ c[2]2(−348c[3]c[4] + 44c[6]) + c[2](−135c[3]3 + 64c[4]2 + 118c[3]c[5]− 19c[7]) + 7c[8])

e8 + . . . .

(19)

Expanding f (ψ2nd NR(x)) about x∗ by Taylor’s method, we have

f (ψ2nd NR(x)) = f ′(x∗)(

c[2]e2 +(− 2c[2]2 + 2c[3]

)e3 +

(5c[2]3 − 7c[2]c[3] + 3c[4]

)e4 − 2

(6c[2]4

− 12c[2]2c[3] + 3c[3]2 + 5c[2]c[4]− 2c[5])

e5 +(

28c[2]5 − 73c[2]3c[3] + 34c[2]2c[4]− 17c[3]c[4]

+ c[2](37c[3]2 − 13c[5]) + 5c[6])

e6 − 2(

32c[2]6 − 103c[2]4c[3]− 9c[3]3 + 52c[2]3c[4] + 6c[4]2

+ c[2]2(80c[3]2 − 22c[5]) + 11c[3]c[5] + c[2](−52c[3]c[4] + 8c[6])− 3c[7])

e7

+(

144c[2]7 − 552c[2]5c[3] + 297c[2]4c[4] + 75c[3]2c[4] + 2c[2]3(291c[3]2 − 67c[5])

− 31c[4]c[5]− 27c[3]c[6] + c[2]2(−455c[3]c[4] + 54c[6]) + c[2](−147c[3]3 + 73c[4]2

+ 134c[3]c[5]− 19c[7]) + 7c[8])

e8 + . . . .)

(20)

Using Equations (17)–(20) in divided difference techniques. We have

f [ψ2nd NR(x), x, x] = f ′(x∗)(

c[2] + 2c[3]e +(

c[2]c[3] + 3c[4])

e2 + 2(− c[2]2c[3] + c[3]2

+ c[2]c[4] + 2c[5])

e3 +(

4c[2]3c[3]− 3c[2]2c[4] + 7c[3]c[4] + c[2](−7c[3]2 + 3c[5]) + 5c[6])

e4

+(− 8c[2]4c[3]− 6c[3]3 + 4c[2]3c[4] + 4c[2]2(5c[3]2 − c[5]) + 10c[3]c[5]

+ 4c[2](−5c[3]c[4] + c[6]) + 6(c[4]2 + c[7]))

e5 +(

16c[2]5c[3]− 4c[2]4c[4]

− 25c[3]2c[4] + 17c[4]c[5] + c[2]3(−52c[3]2 + 5c[5]) + c[2]2(46c[3]c[4]− 5c[6])

+ 13c[3]c[6] + c[2](33c[3]3 − 14c[4]2 − 26c[3]c[5] + 5c[7]) + 7c[8])

e6 + . . . .)

(21)

Substituting Equations (18)–(21) into Equation (7), we obtain, after simplifications,

ψ4thYM(x) = x∗ +(

c[2]3 − c[2]c[3])

e4 − 2(

2c[2]4 − 4c[2]2c[3] + c[3]2 + c[2]c[4])

e5 +(

10c[2]5 − 30c[2]3c[3]

+ 12c[2]2c[4]− 7c[3]c[4] + 3c[2](6c[3]2 − c[5]))

e6 − 2(

10c[2]6 − 40c[2]4c[3]− 6c[3]3

+ 20c[2]3c[4] + 3c[4]2 + 8c[2]2(5c[3]2 − c[5]) + 5c[3]c[5] + c[2](−26c[3]c[4] + 2c[6]))

e7 +(

36c[2]7

− 178c[2]5c[3] + 101c[2]4c[4] + 50c[3]2c[4] + 3c[2]3(84c[3]2 − 17c[5])− 17c[4]c[5]− 13c[3]c[6]

+ c[2]2(−209c[3]c[4] + 20c[6]) + c[2](−91c[3]3 + 37c[4]2 + 68c[3]c[5]− 5c[7]))

e8 + . . . .

(22)

29


Expanding f (ψ4thYM(x)) about x∗ by Taylor’s method, we have

f (ψ4thYM(x)) = f ′(x∗)((

c[2]3 − c[2]c[3])

e4 − 2(

2c[2]4 − 4c[2]2c[3] + c[3]2 + c[2]c[4])

e5 +(

10c[2]5

− 30c[2]3c[3] + 12c[2]2c[4]− 7c[3]c[4] + 3c[2](6c[3]2 − c[5]))

e6 − 2(

10c[2]6 − 40c[2]4c[3]

− 6c[3]3 + 20c[2]3c[4] + 3c[4]2 + 8c[2]2(5c[3]2 − c[5]) + 5c[3]c[5] + c[2](−26c[3]c[4] + 2c[6]))

e7

+(

37c[2]7 − 180c[2]5c[3] + 101c[2]4c[4] + 50c[3]2c[4] + c[2]3(253c[3]2 − 51c[5])− 17c[4]c[5]

− 13c[3]c[6] + c[2]2(−209c[3]c[4] + 20c[6]) + c[2](−91c[3]3 + 37c[4]2 + 68c[3]c[5]− 5c[7]))

e8 + . . . .)

(23)

Now,

f [ψ4thYM(x), x, x] = f ′(x∗)(

c[2] + 2c[3]e + 3c[4]e2 + 4c[5]e3 +(

c[2]3c[3]− c[2]c[3]2 + 5c[6])

e4

+(− 4c[2]4c[3] + 8c[2]2c[3]2 − 2c[3]3 + 2c[2]3c[4]− 4c[2]c[3]c[4] + 6c[7]

)e5

+(

10c[2]5c[3]− 8c[2]4c[4] + 28c[2]2c[3]c[4]− 11c[3]2c[4] + c[2]3(−30c[3]2 + 3c[5])+

2c[2](9c[3]3 − 2c[4]2 − 3c[3]c[5]) + 7c[8])

e6 + . . . .)

(24)

Substituting Equations (19)–(21), (23) and (24) into Equation (12), we obtain, after simplifications,

ψ8thYM(x)− x∗ = c[2]2(

c[2]2 − c[3])(

c[2]3 − c[2]c[3] + c[4])

e8 + O(e9) (25)

Hence from Equations (22) and (25), we concluded that the convergence order of the 4thYM and8thYM are four and eight respectively.

The following theorem is given without proof, which can be worked out with the help of Mathematica.

Theorem 2. Let f : D ⊂ R → R be a sufficiently smooth function having continuous derivatives. If f (x)has a simple root x∗ in the open interval D and x0 chosen in sufficiently small neighborhood of x∗, then themethod (16) is of local sixteenth order convergence and and it satisfies the error equation

ψ16thYM(x)− x∗ =((c[2]4)((c[2]2 − c[3])2)(c[2]3 − c[2]c[3] + c[4])(c[2]4 − c[2]2c[3] + c[2]c[4]− c[5])

)e16 + O(e17).

4. Numerical Examples

In this section, numerical results on some test functions are compared for the new methods4thYM, 8thYM and 16thYM with some existing eighth order methods and Newton’s method. Numericalcomputations have been carried out in the MATLAB software with 500 significant digits. We have usedthe stopping criteria for the iterative process satisfying error = |xN − xN−1| < ε, where ε = 10−50 andN is the number of iterations required for convergence. The computational order of convergence isgiven by ([17])

ρ =ln |(xN − xN−1)/(xN−1 − xN−2)|

ln |(xN−1 − xN−2)/(xN−2 − xN−3)| .

We consider the following iterative methods for solving nonlinear equations for the purpose ofcomparison: ψ4thSB, a method proposed by Sharma et al. [18]:

y = x− 2 f (x)3 f ′(x)

, ψ4thSB(x) = x−(− 1

2+

98

f ′(x)f ′(y) +

38

f ′(y)f ′(x)

) f (x)f ′(x)

. (26)

ψ4thCLND, a method proposed by Chun et al. [19]:

y = x− 2 f (x)3 f ′(x)

, ψ4thCLND(x) = x− 16 f (x) f ′(x)−5 f ′(x)2 + 30 f ′(x) f ′(y)− 9 f ′(y)2 . (27)

30


ψ4thSJ , a method proposed by Singh et al. [20]:

y = x− 23

f (x)f ′(x)

, ψ4thSJ(x) = x−(

178− 9

4f ′(y)f ′(x)

+98

( f ′(y)f ′(x)

)2)(

74− 3

4f ′(y)

f ′(xn)

)f (x)f ′(x)

. (28)

ψ8thKT , a method proposed by Kung-Traub [2]:

y = x− f (x)f ′(x)

, z = y− f (y) ∗ f (x)( f (x)− f (y))2

f (x)f ′(x)

,

ψ8thKT(x) = z− f (x)f ′(x)

f (x) f (y) f (z)( f (x)− f (y))2

f (x)2 + f (y)( f (y)− f (z))( f (x)− f (z))2( f (y)− f (z))

.(29)

ψ8th LW , a method proposed by Liu et al. [8]

y = x− f (x)f ′(x)

, z = y− f (x)f (x)− 2 f (y)

f (y)f ′(x)

,

ψ8th LW(x) = z− f (z)f ′(x)

(( f (x)− f (y)f (x)− 2 f (y)

)2+

f (z)f (y)− f (z)

+4 f (z)

f (x) + f (z)

).

(30)

ψ8thPNPD, a method proposed by Petkovic et al. [11]

y = x− f (x)f ′(x)

, z = x−(( f (y)

f (x)

)2 − f (x)f (y)− f (x)

)f (x)f ′(x)

, ψ8thPNPD(x) = z− f (z)f ′(x)

(ϕ(t) +

f (z)f (y)− f (z)

+4 f (z)f (x)

),

where ϕ(t) = 1 + 2t + 2t2 − t3 and t =f (y)f (x)

.

(31)

ψ8thSA1, a method proposed by Sharma et al. [12]

y = x− f (x)f ′(x)

, z = y−(

3− 2f [y, x]f ′(x)

)f (y)f ′(x)

, ψ8thSA1(x) = z− f (z)f ′(x)

(f ′(x)− f [y, x] + f [z, y]

2 f [z, y]− f [z, x]

). (32)

ψ8thSA2, a method proposed by Sharma et al. [13]

y = x− f (x)f ′(x)

, z = y− f (y)2 f [y, x]− f ′(x)

, ψ8thSA2(x) = z− f [z, y]f [z, x]

f (z)2 f [z, y]− f [z, x]

(33)

ψ8thCFGT , a method proposed by Cordero et al. [6]

y = x− f (x)f ′(x)

, z = y− f (y)f ′(x)

11− 2t + t2 − t3/2

, ψ8thCFGT(x) =

z− 1 + 3r1 + r

f (z)f [z, y] + f [z, x, x](z− y)

, r =f (z)f (x)

.(34)

ψ8thCTV , a method proposed by Cordero et al. [7]

y = x− f (x)f ′(x)

, z = x− 1− t1− 2t

f (x)f ′(x)

, ψ8thCTV(x) = z−( 1− t

1− 2t− v

)2 11− 3v

f (z)f ′(x)

, v =f (z)f (y)

. (35)

Table 1 shows the efficiency indices of the new methods with some known methods.

31


Table 1. Comparison of Efficiency Indices.

Methods p d EI

2ndNR 2 2 1.4144thSB 4 3 1.587

4thCLND 4 3 1.5874thSJ 4 3 1.587

4thYM 4 3 1.5878thKT 8 4 1.6828thLW 8 4 1.682

8thPNPD 8 4 1.6828thSA1 8 4 1.6828thSA2 8 4 1.682

8thCFGT 8 4 1.6828thCTV 8 4 1.6828thYM 8 4 1.68216thYM 16 5 1.741

The following test functions and their simple zeros for our study are given below [10]:

f1(x) = sin(2 cos x)− 1− x2 + esin(x3), x∗ = −0.7848959876612125352...

f2(x) = xex2 − sin2x + 3 cos x + 5, x∗ = −1.2076478271309189270...

f3(x) = x3 + 4x2 − 10, x∗ = 1.3652300134140968457...

f4(x) = sin(x) + cos(x) + x, x∗ = −0.4566247045676308244...

f5(x) =x2− sin x, x∗ = 1.8954942670339809471...

f6(x) = x2 + sin(x5)− 1

4, x∗ = 0.4099920179891371316...

Table 2, shows that corresponding results for f1 − f6. We observe that proposed method 4thYM isconverge in a lesser or equal number of iterations and with least error when compare to comparedmethods. Note that 4thSB and 4thSJ methods are getting diverge in f5 function. Hence, the proposedmethod 4thYM can be considered competent enough to existing other compared equivalent methods.

Also, from Tables 3–5 are shows the corresponding results for f1 − f6. The computational order ofconvergence agrees with the theoretical order of convergence in all the functions. Note that 8thPNPDmethod is getting diverge in f5 function and all other compared methods are converges with least error.Also, function f1 having least error in 8thCFGT, function f2 having least error in 8thCTV, functions f3

and f4 having least error in 8thYM, function f5 having least error in 8thSA2, function f6 having leasterror in 8thCFGT. The proposed 16thYM method converges less number of iteration with least errorin all the tested functions. Hence, the 16thYM can be considered competent enough to existing othercompared equivalent methods.

32


Table 2. Numerical results for nonlinear equations.

Methods f1(x), x0 = −0.9 f2(x), x0 = −1.6

N |x1 − x0| |xN − xN−1| ρ N |x1 − x0| |xN − xN−1| ρ

2nd NR (4) 7 0.1080 7.7326 × 10−74 1.99 9 0.2044 9.2727 × 10−74 1.994thSB (26) 4 0.1150 9.7275 × 10−64 3.99 5 0.3343 1.4237 × 10−65 3.99

4thCLND (27) 4 0.1150 1.4296 × 10−64 3.99 5 0.3801 1.1080 × 10−72 3.994thSJ (28) 4 0.1150 3.0653 × 10−62 3.99 5 0.3190 9.9781 × 10−56 3.994thYM (7) 4 0.1150 6.0046 × 10−67 3.99 5 0.3737 7.2910 × 10−120 4.00

Methods f3(x), x0 = 0.9 f4(x), x0 = −1.9

2nd NR (4) 8 0.6263 1.3514 × 10−72 2.00 8 1.9529 1.6092 × 10−72 1.994thSB (26) 5 0.5018 4.5722 × 10−106 3.99 5 1.5940 6.0381 × 10−92 3.99

4thCLND (27) 5 0.5012 4.7331 × 10−108 3.99 5 1.5894 2.7352 × 10−93 3.994thSJ (28) 5 0.4767 3.0351 × 10−135 3.99 5 1.5776 9.5025 × 10−95 3.994thYM (7) 5 0.4735 2.6396 × 10−156 3.99 5 1.5519 1.4400 × 10−102 3.99

Methods f5(x), x0 = 1.2 f6(x), x0 = 0.8

2nd NR (4) 9 2.4123 1.3564 × 10−83 1.99 8 0.3056 3.2094 × 10−72 1.994thSB (26) Diverge 5 0.3801 2.8269 × 10−122 3.99

4thCLND (27) 14 0.0566 6.8760 × 10−134 3.99 5 0.3812 7.8638 × 10−127 3.994thSJ (28) Diverge 5 0.3780 1.4355 × 10−114 3.994thYM (7) 6 1.2887 2.3155 × 10−149 3.99 5 0.3840 1.1319 × 10−143 3.99


Methods f1(x), x0 = −0.9 f2(x), x0 = −1.6


8thKT (29) 3 0.1151 1.6238 × 10−61 7.91 4 0.3876 7.2890 × 10−137 7.998thLW (30) 3 0.1151 4.5242 × 10−59 7.91 4 0.3904 1.1195 × 10−170 8.00

8thPNPD (31) 3 0.1151 8.8549 × 10−56 7.87 4 0.3734 2.3461 × 10−85 7.998thSA1 (32) 3 0.1151 3.4432 × 10−60 7.88 4 0.3983 8.4343 × 10−121 8.008thSA2 (33) 3 0.1151 6.9371 × 10−67 7.99 4 0.3927 5.9247 × 10−225 7.99

8thCFGT (34) 3 0.1151 1.1715 × 10−82 7.77 5 0.1532 2.0650 × 10−183 7.998thCTV (35) 3 0.1151 4.4923 × 10−61 7.94 4 0.3925 2.3865 × 10−252 7.998thYM (12) 3 0.1151 1.1416 × 10−70 7.96 4 0.3896 8.9301 × 10−163 8.0016thYM (16) 3 0.1151 0 15.99 3 0.3923 3.5535 × 10−85 16.20


Methods f3(x), x0 = 0.9 f4(x), x0 = −1.9


8thKT (29) 4 0.4659 5.0765 × 10−216 7.99 4 1.4461 5.5095 × 10−204 8.008thLW (30) 4 0.4660 2.7346 × 10−213 7.99 4 1.4620 3.7210 × 10−146 8.00

8thPNPD (31) 4 0.3821 9.9119 × 10−71 8.02 4 1.3858 2.0603 × 10−116 7.988thSA1 (32) 4 0.4492 1.5396 × 10−122 8.00 4 1.4170 2.2735 × 10−136 7.998thSA2 (33) 4 0.4652 4.1445 × 10−254 7.98 4 1.4339 2.5430 × 10−166 7.99

8thCFGT (34) 4 0.4654 2.4091 × 10−260 7.99 4 1.4417 4.7007 × 10−224 7.998thCTV (35) 4 0.4652 3.8782 × 10−288 8.00 4 1.3957 3.7790 × 10−117 7.998thYM (12) 4 0.4653 3.5460 × 10−309 7.99 4 1.4417 2.9317 × 10−229 7.9916thYM (16) 3 0.4652 3.6310 × 10−154 16.13 3 1.4434 1.8489 × 10−110 16.36

33



Methods f5(x), x0 = 1.2 f6(x), x0 = 0.8


8thKT (29) 5 1.8787 2.6836 × 10−182 7.99 4 0.3898 6.0701 × 10−234 7.998thLW (30) 6 40.5156 4.6640 × 10−161 7.99 4 0.3898 6.1410 × 10−228 7.99

8thPNPD (31) Diverge 4 0.3894 3.6051 × 10−190 7.998thSA1 (32) 7 891.9802 2.1076 × 10−215 9.00 4 0.3901 5.9608 × 10−245 8.008thSA2 (33) 4 0.7161 5.3670 × 10−128 7.99 4 0.3900 8.3398 × 10−251 8.61

8thCFGT (34) 5 2.8541 0 7.99 4 0.3900 0 7.998thCTV (35) 5 0.6192 1.6474 × 10−219 9.00 4 0.3901 1.0314 × 10−274 8.008thYM (12) 4 0.7733 1.3183 × 10−87 7.98 4 0.3900 1.2160 × 10−286 7.9916thYM (16) 4 0.6985 0 16.10 3 0.3900 1.1066 × 10−143 15.73

5. Applications to Some Real World Problems

5.1. Projectile Motion Problem

We consider the classical projectile problem [21,22] in which a projectile is launched from a towerof height h > 0, with initial speed v and at an angle θ with respect to the horizontal onto a hill, whichis defined by the function ω, called the impact function which is dependent on the horizontal distance,x. We wish to find the optimal launch angle θm which maximizes the horizontal distance. In ourcalculations, we neglect air resistances.

The path function y = P(x) that describes the motion of the projectile is given by

P(x) = h + x tan θ − gx2

2v2 sec2 θ (36)

When the projectile hits the hill, there is a value of x for which P(x) = ω(x) for each value of x.We wish to find the value of θ that maximize x.

ω(x) = P(x) = h + x tan θ − gx2

2v2 sec2 θ (37)

Differentiating Equation (37) implicitly w.r.t. θ, we have

ω′(x)dxdθ

= x sec2 θ +dxdθ

tan θ − gv2

(x2 sec2 θ tan θ + x

dxdθ

sec2 θ

)(38)

Settingdxdθ

= 0 in Equation (38), we have

xm =v2

gcot θm (39)

or

θm = arctan(

v2

g xm

)(40)

An enveloping parabola is a path that encloses and intersects all possible paths. Henelsmith [23]derived an enveloping parabola by maximizing the height of the projectile fora given horizontal distancex, which will give the path that encloses all possible paths. Let w = tan θ, then Equation (36) becomes

y = P(x) = h + xw− gx2

2v2 (1 + w2) (41)

34


Differentiating Equation (41) w.r.t. w and setting y′ = 0, Henelsmith obtained

y′ = x− xg2

v2 (w) = 0

w =v2

g x

(42)

so that the enveloping parabola defined by

ym = ρ(x) = h +v2

2g− gx2

2v2 (43)

The solution to the projectile problem requires first finding xm which satisfies ρ(x) = ω(x) andsolving for θm in Equation (40) because we want to find the point at which the enveloping parabolaρ intersects the impact function ω, and then find θ that corresponds to this point on the envelopingparabola. We choose a linear impact function ω(x) = 0.4x with h = 10 and v = 20. We let g = 9.8.Then we apply our IFs starting from x0 = 30 to solve the non-linear equation

f (x) = ρ(x)−ω(x) = h +v2

2g− gx2

2v2 − 0.4x

whose root is given by xm = 36.102990117..... and

θm = arctan(

v2

g xm

)= 48.5◦.

Figure 1 shows the intersection of the path function, the enveloping parabola and the linear impactfunction for this application. The approximate solutions are calculated correct to 500 significant figures.The stopping criterion |xN − xN−1| < ε, where ε = 10−50 is used. Table 6 shows that proposed method16thYM is converging better than other compared methods. Also, we observe that the computationalorder of convergence agrees with the theoretical order of convergence.

0 10 20 30 40 50−5

0

5

10

15

20

25

30

35

P(x)ω(x)ρ(x)

Figure 1. The enveloping parabola with linear impact function.

35


Table 6. Results of projectile problem.

IF N Error cpu Time(s) ρ

2ndNR 7 4.3980 × 10−76 1.074036 1.994thYM 4 4.3980 × 10−76 0.902015 3.998thKT 3 1.5610 × 10−66 0.658235 8.038thLW 3 7.8416 × 10−66 0.672524 8.03

8thPNPD 3 4.2702 × 10−57 0.672042 8.058thSA1 3 1.2092 × 10−61 0.654623 8.068thCTV 3 3.5871 × 10−73 0.689627 8.028thYM 3 4.3980 × 10−76 0.618145 8.0216thYM 3 0 0.512152 16.01

5.2. Planck’s Radiation Law Problem

We consider the following Planck’s radiation law problem found in [24]:

ϕ(λ) =8πchλ−5

ech/λkT − 1, (44)

which calculates the energy density within an isothermal blackbody. Here, λ is the wavelength ofthe radiation, T is the absolute temperature of the blackbody, k is Boltzmann’s constant, h is thePlanck’s constant and c is the speed of light. Suppose, we would like to determine wavelength λ whichcorresponds to maximum energy density ϕ(λ). From (44), we get

ϕ′(λ) =( 8πchλ−6

ech/λkT − 1

)( (ch/λkT)ech/λkT

ech/λkT − 1− 5

)= A · B.

It can be checked that a maxima for ϕ occurs when B = 0, that is, when

( (ch/λkT)ech/λkT

ech/λkT − 1

)= 5.

Here putting x = ch/λkT, the above equation becomes

1− x5= e−x. (45)

Definef (x) = e−x − 1 +

x5

. (46)

The aim is to find a root of the equation f (x) = 0. Obviously, one of the root x = 0 is not taken fordiscussion. As argued in [24], the left-hand side of (45) is zero for x = 5 and e−5 ≈ 6.74× 10−3. Hence,it is expected that another root of the equation f (x) = 0 might occur near x = 5. The approximateroot of the Equation (46) is given by x∗ ≈ 4.96511423174427630369 with x0 = 3. Consequently, thewavelength of radiation (λ) corresponding to which the energy density is maximum is approximated as

λ ≈ ch(kT)4.96511423174427630369

.

Table 7 shows that proposed method 16thYM is converging better than other compared methods.Also, we observe that the computational order of convergence agrees with the theoretical orderof convergence.

36


Table 7. Results of Planck’s radiation law problem.

IF N Error cpu Time(s) ρ

2ndNR 7 1.8205 × 10−70 0.991020 2.004thYM 5 1.4688 × 10−181 0.842220 4.008thKT 4 4.0810 × 10−288 0.808787 7.998thLW 4 3.1188 × 10−268 0.801304 7.99

8thPNPD 4 8.0615 × 10−260 0.800895 7.998thSA1 4 1.9335 × 10−298 0.791706 8.008thCTV 4 5.8673 × 10−282 0.831006 8.008thYM 4 2.5197 × 10−322 0.855137 8.00

16thYM 3 8.3176 × 10−153 0.828053 16.52

Hereafter, we will study the optimal fourth and eighth order methods along with Newton’s method.

6. Corresponding Conjugacy Maps for Quadratic Polynomials

In this section, we discuss the rational map Rp arising from 2ndNR, proposed methods 4thYMand 8thYM applied to a generic polynomial with simple roots.

Theorem 3. (2ndNR) [18] For a rational map Rp(z) arising from Newton’s method (4) applied to p(z) =

(z− a)(z− b), a �= b, Rp(z) is conjugate via the a Möbius transformation given by M(z) = (z− a)/(z− b) to

S(z) = z2.

Theorem 4. (4thYM) For a rational map Rp(z) arising from Proposed Method (7) applied to p(z) = (z−a)(z− b), a �= b, Rp(z) is conjugate via the a Möbius transformation given by M(z) = (z− a)/(z− b) to

S(z) = z4.

Proof. Let p(z) = (z − a)(z − b), a �= b, and let M be Möbius transformation given by M(z) =

(z− a)/(z− b) with its inverse M−1(z) = (zb−a)(z−1) , which may be considered as map from C ∪ {∞}.

We then haveS(z) = M ◦ Rp ◦M−1(z) = M

(Rp

( zb− az− 1

))= z4.

Theorem 5. (8thYM) For a rational map Rp(z) arising from Proposed Method (12) applied to p(z) = (z−a)(z− b), a �= b, Rp(z) is conjugate via the a Möbius transformation given by M(z) = (z− a)/(z− b) to

S(z) = z8.

Proof. Let p(z) = (z − a)(z − b), a �= b, and let M be Möbius transformation given by M(z) =

(z− a)/(z− b) with its inverse M−1(z) = (zb−a)(z−1) , which may be considered as map from C ∪ {∞}.

We then haveS(z) = M ◦ Rp ◦M−1(z) = M

(Rp

( zb− az− 1

))= z8.

Remark 1. The methods (29)–(35) are given without proof, which can be worked out with the help of Mathematica.

Remark 2. All the maps obtained above are of the form S(z) = zpR(z), where R(z) is either unity or a rationalfunction and p is the order of the method.

37


7. Basins of Attraction

The study of dynamical behavior of the rational function associated to an iterative method givesimportant information about convergence and stability of the method. The basic definitions anddynamical concepts of rational function which can found in [4,25].

We take a square R×R = [−2, 2]× [−2, 2] of 256× 256 points and we apply our iterative methodsstarting in every z(0) in the square. If the sequence generated by the iterative method attempts a zeroz∗j of the polynomial with a tolerance | f (z(k))| < ×10−4 and a maximum of 100 iterations, we decide

that z(0) is in the basin of attraction of this zero. If the iterative method starting in z(0) reaches a zero inN iterations (N ≤ 100), then we mark this point z(0) with colors if |z(N) − z∗j | < ×10−4. If N > 50, weconclude that the starting point has diverged and we assign a dark blue color. Let ND be a numberof diverging points and we count the number of starting points which converge in 1, 2, 3, 4, 5 orabove 5 iterations. In the following, we describe the basins of attraction for Newton’s method andsome higher order Newton type methods for finding complex roots of polynomials p1(z) = z2 − 1,p2(z) = z3 − 1 and p3(z) = z5 − 1.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

p1(z) = z2 − 1 p2(z) = z3 − 1 p3(z) = z5 − 1

Figure 2. Basins of attraction for 2nd NR for the polynomial p1(z), p2(z), p3(z).

Figures 2 and 3 shows the polynomiographs of the methods for the polynomial p1(z). We cansee that the methods 2ndNR, 4thYM, 8thSA2 and 8thYM performed very nicely. The methods 4thSB,4thSJ, 8thKT, 8thLW, 8thPNPD, 8thSA1, 8thCFGT and 8thCTV are shows some chaotic behavior near theboundary points. The method 4thCLND have sensitive to the choice of initial guess in this case.

Figures 2 and 4 shows the polynomiographs of the methods for the polynomial p2(z). We can seethat the methods 2ndNR, 4thYM, 8thSA2 and 8thYM performed very nicely. The methods 4thSB, 8thKT,8thLW and 8thCTV are shows some chaotic behavior near the boundary points. The methods 4thCLND,4thSJ, 8thPNPD, 8thSA1, and 8thCFGT have sensitive to the choice of initial guess in this case.

Figures 2 and 5 shows the polynomiographs of the methods for the polynomial p3(z). We cansee that the methods 4thYM, 8thSA2 and 8thYM are shows some chaotic behavior near the boundarypoints. The methods 2ndNR, 4thSB, 4thCLND, 4thSJ, 8thKT, 8thLW, 8thPNPD, 8thSA1, 8thCFGT and8thCTV have sensitive to the choice of initial guess in this case. In Tables 8–10, we classify the numberof converging and diverging grid points for each iterative method.

38


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thSB 4thCLND 4thSJ

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thYM 8thKT 8thLW

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

8thPNPD 8thSA1 8thSA2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

8thCFGT 8thCTV 8thYM

Figure 3. Basins of attraction for p1(z) = z2 − 1.

39


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thSB 8thCLND 8thSJ

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thYM 8thKT 8thLW

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2



40


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thSB 4thCLND 4thSJ

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

4thYM 8thKT 8thLW

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2



41


Table 8. Results of the polynomials p1(z) = z2 − 1.

IF N = 1 N = 2 N = 3 N = 4 N = 5 N > 5 ND

2ndNR 4 516 7828 23,272 20,548 13,368 04thSB 340 22,784 29,056 6836 2928 3592 0

4thCLND 372 24,600 29,140 6512 2224 2688 10764thSJ 300 19,816 28,008 5844 2968 8600 0

4thYM 520 31,100 27,520 4828 1208 360 08thKT 4684 44,528 9840 3820 1408 1256 248thLW 4452 43,236 11,408 3520 1540 1380 0

8thPNPD 2732 39,768 13,112 3480 1568 4876 168thSA1 4328 45,824 8136 2564 1484 3200 08thSA2 15,680 45,784 3696 376 0 0 0

8thCFGT 9616 43,716 7744 2916 980 564 648thCTV 7124 48,232 7464 1892 632 192 08thYM 8348 50,792 5572 824 0 0 0


IF N = 1 N = 2 N = 3 N = 4 N = 5 N > 5 ND

2ndNR 0 224 2908 11,302 19,170 31,932 04thSB 160 9816 27,438 9346 5452 13,324 6

4thCLND 170 11,242 28,610 9984 4202 11,328 71764thSJ 138 7760 25,092 8260 5058 19,228 1576

4thYM 270 18,064 30,374 9862 3688 3278 08thKT 2066 34,248 11,752 6130 4478 6862 08thLW 2092 33,968 12,180 4830 3030 9436 0

8thPNPD 1106 25,712 11,258 3854 1906 21,700 10,4528thSA1 1608 36,488 12,486 3718 1780 9456 8728thSA2 6432 46,850 9120 2230 640 264 0

8thCFGT 3688 40,740 13,696 4278 1390 1744 73958thCTV 3530 43,554 11,724 3220 1412 2096 08thYM 3816 43,596 12,464 3636 1302 722 0


IF N = 1 N = 2 N = 3 N = 4 N = 5 N > 5 ND

2ndNR 2 100 1222 4106 7918 52,188 6384thSB 76 3850 15,458 18,026 5532 22,594 5324

4thCLND 86 4476 18,150 17,774 5434 19,616 12,2084thSJ 62 3094 11,716 16,840 5682 28,142 19,900

4thYM 142 7956 27,428 15,850 5726 8434 08thKT 950 17,884 20,892 5675 4024 16,111 2178thLW 1032 18,764 20,622 5056 3446 16,616 1684

8thPNPD 496 12,770 21,472 6576 2434 21,788 14,2368thSA1 692 26,212 15,024 4060 1834 17,714 88148thSA2 2662 41,400 12,914 4364 1892 2304 0

8thCFGT 2008 21,194 23,734 6180 3958 8462 19538thCTV 1802 36,630 13,222 4112 2096 7674 3508thYM 1736 27,808 21,136 5804 2704 6348 0

We note that a point z0 belongs to the Julia set if and only if the dynamics in a neighborhood ofz0 displays sensitive dependence on the initial conditions, so that nearby initial conditions lead towildly different behavior after a number of iterations. For this reason, some of the methods are gettingdivergent points. The common boundaries of these basins of attraction constitute the Julia set of the

42


iteration function. It is clear that one has to use quantitative measures to distinguish between themethods, since we have a different conclusion when just viewing the basins of attraction.

In order to summarize the results, we have compared mean number of iteration and total numberof functional evaluations (TNFE) for each polynomials and each methods in Table 11. The best methodbased on the comparison in Table 11 is 8thSA2. The method with the fewest number of functionalevaluations per point is 8thSA2 followed closely by 4thYM. The fastest method is 8thSA2 followedclosely by 8thYM. The method with highest number of functional evaluation and slowest methodis 8thPNPD.

Table 11. Mean number of iteration (Nμ) and TNFE for each polynomials and each methods.

IF Nμ f or p1(z) Nμ f or p2(z) Nμ f or p3(z) Average TNFE

2ndNR 4.7767 6.4317 9.8531 7.0205 14.04104thSB 3.0701 4.5733 9.2701 5.6378 16.9135

4thCLND 3.6644 8.6354 12.8612 8.3870 25.16104thSJ 3.7002 7.0909 14.5650 8.4520 25.3561

4thYM 2.6366 3.1733 4.0183 3.2760 9.82828thKT 2.3647 3.1270 4.4501 3.3139 13.25578thLW 2.3879 3.5209 6.3296 4.0794 16.3178

8thPNPD 2.9959 10.5024 12.3360 8.6114 34.44578thSA1 2.5097 4.5787 9.7899 5.6262 22.50448thSA2 1.8286 2.1559 2.5732 2.1859 8.7436

8thCFGT 2.1683 2.8029 3.4959 2.8223 11.28948thCTV 2.1047 2.4708 3.9573 2.8442 11.37708thYM 1.9828 2.3532 3.3617 2.5659 10.2636

8. Concluding Remarks and Future Work

In this work, we have developed optimal fourth, eighth and sixteenth order iterative methodsfor solving nonlinear equations using the divided difference approximation. The methods requirethe computations of three functions evaluations reaching order of convergence is four, four functionsevaluations reaching order of convergence is eight and five functions evaluations reaching order ofconvergence is sixteen. In the sense of convergence analysis and numerical examples, the Kung-Traub’sconjecture is satisfied. We have tested some examples using the proposed schemes and some knownschemes, which illustrate the superiority of the proposed method 16thYM. Also, proposed methodsand some existing methods have been applied on the Projectile motion problem and Planck’s radiationlaw problem. The results obtained are interesting and encouraging for the new method 16thYM.The numerical experiments suggests that the new methods would be valuable alternative for solvingnonlinear equations. Finally, we have also compared the basins of attraction of various fourth andeighth order methods in the complex plane.

Future work includes:

• Now we are investigating the proposed scheme to develop optimal methods of arbitrarily highorder with Newton’s method, as in [26].

• Also, we are investigating to develop derivative free methods to study dynamical behavior andlocal convergence, as in [27,28].

Author Contributions: The contributions of both the authors have been similar. Both of them have workedtogether to develop the present manuscript.

Funding: This paper is supported by three project funds: 1. National College Students Innovation andentrepreneurship training program of Ministry of Education of the People’s Republic of China in 2017: InternetAnimation Company in Minority Areas–Research Model of “Building Dream” Animation Company. (Projectnumber: 201710684001). 2. Yunnan Provincial Science and Technology Plan Project University Joint Project2017: Research on Boolean Satisfiability Dividing and Judging Method Based on Clustering and Partitioning(Project number: 2017FH001-056). 3. Qujing Normal college scientific research fund special project (Projectnumber: 2018zx003).

43


Acknowledgments: The authors would like to thank the editors and referees for the valuable comments and forthe suggestions to improve the readability of the paper.


References

1. Ostrowski, A.M. Solutions of Equations and System of Equations; Academic Press: New York, NY, USA, 1960.2. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 1974, 21,

643–651.3. Amat, S.; Busquier, S.; Plaza, S. Dynamics of a family of third-order iterative methods that do not require

using second derivatives. Appl. Math. Comput. 2004, 154, 735–746.4. Amat, S.; Busquier, S.; Plaza, S. Review of some iterative root-finding methods from a dynamical point of

view. Scientia 2004, 10, 3–35.5. Babajee, D.K.R; Madhu, K.; Jayaraman, J. A family of higher order multi-point iterative methods based on

power mean for solving nonlinear equations. Afrika Matematika 2016, 27, 865–876. [CrossRef]6. Cordero, A.; Fardi, M.; Ghasemi, M.; Torregrosa, J.R. Accelerated iterative methods for finding solutions of

nonlinear equations and their dynamical behavior. Calcolo 2014, 51, 17–30.7. Cordero, A.; Torregrosa, J.R.; Vasileva, M.P. A family of modified ostrowski’s methods with optimal eighth

order of convergence. Appl. Math. Lett. 2011, 24, 2082–2086.8. Liu, L.; Wang, X. Eighth-order methods with high efficiency index for solving nonlinear equations.

Appl. Math. Comput. 2010, 215, 3449–3454.9. Madhu, K. Some New Higher Order Multi-Point Iterative Methods and Their Applications to Differential

and Integral Equation and Global Positioning System. Ph.D. Thesis, Pndicherry University, Kalapet, India,June 2016.

10. Madhu, K.; Jayaraman, J. Higher order methods for nonlinear equations and their basins of attraction.Mathematics 2016, 4, 22.

11. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Dzunic, J. Multipoint Methods for Solving Nonlinear Equations; Elsevier:Amsterdam, The Netherlands, 2012.

12. Sharma, J.R.; Arora, H. An efficient family of weighted-newton methods with optimal eighth orderconvergence. Appl. Math. Lett. 2014, 29, 1–6.

13. Sharma, J.R.; Arora, H. A new family of optimal eighth order methods with dynamics for nonlinear equations.Appl. Math. Comput. 2016, 273, 924–933.

14. Soleymani, F.; Khratti, S.K.; Vanani, S.K. Two new classes of optimal Jarratt-type fourth-order methods.Appl. Math. Lett. 2011, 25, 847–853.

15. Wait, R. The Numerical Solution of Algebraic Equations; John Wiley and Sons: Hoboken, NJ, USA, 1979.16. Khan, Y.; Fardi, M.; Sayevand, K. A new general eighth-order family of iterative methods for solving

nonlinear equations. Appl. Math. Lett. 2012, 25, 2262–2266.17. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas.

Appl. Math. Comput. 2007, 190, 686–698.18. Sharma, R.; Bahl, A. An optimal fourth order iterative method for solving nonlinear equations and its

dynamics. J. Complex Anal. 2015, 2015, 259167.19. Chun, C; Lee, M. Y.; Neta, B; Dzunic, J. On optimal fourth-order iterative methods free from second derivative

and their dynamics. Appl. Math. Comput. 2012, 218, 6427–6438. [CrossRef]20. Singh, A.; Jaiswal, J.P. Several new third-order and fourth-order iterative methods for solving nonlinear

equations. Int. J. Eng. Math. 2014, 2014, 828409.21. Babajee, D.K.R.; Madhu, K. Comparing two techniques for developing higher order two-point iterative

methods for solving quadratic equations. SeMA J. 2018, 1–22. [CrossRef]22. Kantrowitz, R.; Neumann, M.M. Some real analysis behind optimization of projectile motion.

Mediterr. J. Math. 2014, 11, 1081–1097.23. Henelsmith, N. Finding the Optimal Launch Angle; Whitman College: Walla Walla, WA, USA, 2016.24. Bradie, B. A Friendly Introduction to Numerical Analysis; Pearson Education Inc.: New Delhi, India, 2006.25. Scott, M.; Neta, B.; Chun, C. Basin attractors for various methods. Appl. Math. Comput. 2011, 218, 2584–2599.

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26. Cordero, A.; Hueso, J. L.; Martinez, E.; Torregrosa, J.R. Generating optimal derivative free iterative methodsfor nonlinear equations by using polynomial interpolation. Appl. Math. Comput. 2013, 57 1950–1956.[CrossRef]

27. Argyros, I.K.; Magrenan, A.A.; Orcos, L. Local convergence and a chemical application of derivative freeroot finding methods with one parameter based on interpolation. J. Math. Chem. 2016, 54, 1404–1416.

28. Zafar, F.; Cordero, A.; Torregrosa, J.R. An efficient family of optimal eighth-order multiple root finders.Mathematics 2018, 6, 310.


45

mathematics

Article

Design and Complex Dynamics of Potra–Pták-TypeOptimal Methods for Solving Nonlinear Equationsand Its Applications

Prem B. Chand 1,†, Francisco I. Chicharro 2,† , Neus Garrido 3,*,† and Pankaj Jain 1,†

1 Department of Mathematics, South Asian University, Akbar Bhawan, Chanakya Puri, New Delhi 110021,India; [email protected] (P.B.C.); [email protected] (P.J.)

2 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av. La Paz 137,26006 Logroño, Spain; [email protected]

3 Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, Cno. de Vera s/n,46022 València, Spain

* Correspondence: [email protected]† These authors contributed equally to this work.

Received: 10 September 2019; Accepted: 9 October 2019; Published: 11 October 2019

Abstract: In this paper, using the idea of weight functions on the Potra–Pták method, an optimalfourth order method, a non optimal sixth order method, and a family of optimal eighth order methodsare proposed. These methods are tested on some numerical examples, and the results are comparedwith some known methods of the corresponding order. It is proved that the results obtained from theproposed methods are compatible with other methods. The proposed methods are tested on someproblems related to engineering and science. Furthermore, applying these methods on quadratic andcubic polynomials, their stability is analyzed by means of their basins of attraction.

Keywords: nonlinear equations; Potra–Pták method; optimal methods; weight function; basin ofattraction; engineering applications

PACS: 65H05

1. Introduction

For solving nonlinear equations iteratively, the Newton’s method given by

xn+1 = xn − f (xn)

f ′(xn)

is one of the most commonly used methods. The efficiency index as defined by Ostroswki in [1],which relates the order of convergence of a method p with the number of function evaluations periteration d, is given by the expression p1/d. Newton’s method is quadratically convergent and requirestwo function evaluations per iteration and, thereby, has the efficiency index value of 21/2 ≈ 1.414.Numerous methods have appeared giving higher order of convergence or better efficiency. One ofthe recent strategies to increase the order of the methods is the use of weight functions [2–5]. In thisregard, Sharma and Behl [6] presented the fourth order method:

yn = xn − 23

f (xn)f ′(xn)

,

xn+1 = xn −(− 1

2 + 38

f ′(yn)f ′(xn)

+ 98

f ′(xn)f ′(yn)

)f (xn)f ′(xn)

.(1)



Similarly, Sharifi et al. [7] used weight functions on the third order Heun’s method and proposed thefourth order method

yn = xn − 23

f (xn)f ′(xn)

,

xn+1 = xn − f (xn)4

(1

f ′(xn)+ 3

f ′(yn)

)(1 + 3

8

(f ′(yn)f ′(xn)

− 1)2 − 69

64

(f ′(yn)f ′(xn)

− 1)3

+ f (xn)f ′(yn)

).

(2)

According to Kung and Traub [8], an iterative method is said to be optimal if its order is 2d−1, where dis the number of function evaluations per iteration. Notice that Newton’s method as well as (1) and (2)are all optimal.

Potra and Pták [9], as an attempt to improve Newton’s method, gave the method

yn = xn − f (xn)f ′(xn)

,

xn+1 = xn − f (xn)+ f (yn)f ′(xn)

.(3)

This method is cubically convergent but is not optimal, as it requires three function evaluationsper iteration.

The aim, in the present paper, is to further investigate the method (3). Precisely, we use weightfunctions and improve the order of convergence of (3). We do it in three ways which correspond to themethods of orders 4, 6 and 8. Out of these, the methods with orders 4 and 8 are optimal.

Dynamics of a rational operator give important information about the convergence, efficiencyand stability of the iterative methods. During the last few decades, many researchers, e.g., [10–16]and references therein, study the dynamical behavior of rational operators associated with iterativemethods. Furthermore, there is an extensive literature [17–21] to understand and implement furtherresults on the dynamics of rational functions. In this paper, we also analyze the dynamical behaviorof the methods that we have developed in this paper. Furthermore, at the end of this work, thebasins of attraction are also presented and compared among the proposed and other methods of thecorresponding order.

The remaining part of the paper is organized as follows. In Section 2, the development of themethods and their convergence analysis are given. In Section 3, the proposed methods are tested onsome functions, and the results are compared with other methods in the head of Numerical Examples.In Section 4, the proposed methods are tested on some engineering and science related designs.Section 5 is devoted to analyze the stability of the introduced methods by means of complex dynamics.In this sense, the study of the rational function resulting from the application of the methods to severalnonlinear functions is developed, and their basins of attraction are represented. Finally, Section 6covers the conclusions of the research.

2. Development of Methods and Their Convergence Analysis

In this section, the methods of order four, six and eight are introduced, and its convergenceis analyzed.

2.1. Optimal Fourth Order Method

Based on the Potra–Pták method (3), we propose the following two-step method using a weightfunction, whose iterative expression is


,

xn+1 = xn − w(tn)f (xn)+ f (yn)

f ′(xn),

(4)

where w(tn) = a1 + a2tn + a3t2n and tn = f (yn)

f (xn). The convergence of (4) is proved in the

following theorem.

47


Theorem 1. Let f be a real or complex valued function defined in the interval I having a sufficient number ofsmooth derivatives. Let α be a simple root of the equation f (x) = 0 and the initial point x0 is close enough to α.Then, the method (4) is fourth order of convergence if a1 = 1, a2 = 0 and a3 = 2.

Proof. We denote cj =f (j)(α)j! f ′(α) . Let en = xn − α be the error in xn. Then, Taylor’s series expansion of

f (xn) and f ′(xn) about α gives

f (xn) = f ′(α)(

en + c2e2n + c3e3

n + c4e4n + c5e5

n + c6e6n + c7e7

n + c8e8n + O(e9

n))

(5)

andf ′(xn) = f ′(α)

(1 + 2c2en + 3c3e2

n + 4c4e3n + 5c5e4

n + 6c6e5n + 7c7e6

n + 8c8e7n + O(e8

n))

. (6)

Let dn = yn − α, then, from the first equation of (4), we get

dn = c2e2n − 2

(c2

2 − c3)

e3n +

(4c3

2 − 7c2c3 + 3c4)

e4n +

(−8c42 + 20c3c2

2 − 10c4c2 − 6c23 + 4c5

)e5

n+(16c5

2 − 52c3c32 + 28c4c2

2 +(33c2

3 − 13c5)

c2 − 17c3c4 + 5c6)

e6n

−2(

16c62 − 64c3c4

2 + 36c4c32 + 9

(7c2

3 − 2c5)

c22 + (8c6 − 46c3c4) c2 − 9c3

3 + 6c24 + 11c3c5 − 3c7

)e7

n

+

(64c7

2 − 304c3c52 + 176c4c4

2 +(408c2

3 − 92c5)

c32 + (44c6 − 348c3c4) c2

2

+75c23c4 − 31c4c5 − 27c3c6 + c2

(−135c33 + 118c5c3 + 64c2

4 − 19c7)+ 7c8

)e8

n + O(e9n)

so that, using Taylor’s series expansion of f (yn) about α, we get

f (yn) = f (dn + α)

= f ′(α)[

c2e2n − 2(c2

2 − c3)e3n +

(5c3

2 − 7c2c3 + 3c4)

e4n − 2

(6c4

2 − 12c3c22 + 5c4c2 + 3c2

3 − 2c5)

e5n

+(

28c52 − 73c3c3

2 + 34c4c22 +

(37c2

3 − 13c5)

c2 − 17c3c4 + 5c6

)e6

n

−2(

32c62 − 103c3c4

2 + 52c4c32 +

(80c2

3 − 22c5)

c22 + (8c6 − 52c3c4) c2

−9c33 + 6c2

4 + 11c3c5 − 3c7

)e7

n +

(144c7

2 − 552c3c52 + 297c4c4

2 + 2(291c2

3 − 67c5)

c32

+ (54c6 − 455c3c4) c22 + 75c2

3c4 − 31c4c5 − 27c3c6 + c2(−147c3

3 + 134c5c3 + 73c24 − 19c7

)+7c8

)e8

n + O(e9n)].

(7)

Now, from (5) and (7), we get

tn =f (yn)f (xn)

= c2en + (−3c22 + 2c3)e2

n + (8c32 − 10c2c3 + 3c4)e3

n +(−20c4

2 + 37c22c3 − 14c2c4 − 8c2

3 + 4c5)

e4n

+(48c5

2 − 118c32c3 + 51c2

2c4 + 55c2c23 − 18c2c5 − 22c3c4 + 5c6

)e5

n

+

(− 112c6

2 + 344c3c42 − 163c4c3

2 +(65c5 − 252c2

3)

c22 + 2 (75c3c4 − 11c6) c2

+26c33 − 15c2

4 − 28c3c5 + 6c7

)e6

n + O(e7n).

(8)

Therefore, using the results obtained above in the second equation of (4), we get

en+1 = (1− a1) en − a2c2e2n +

(2a1c2

2 + 3a2c22 − 2a2c3 − a3c2

2)

e3n

+(−9a1c3

2 + 7a1c2c3 − 6a2c32 + 10a2c2c3 − 3a2c4 + 6a3c3

2 − 4a3c2c3)

e4n + O(e5

n).(9)

48


In order to obtain fourth order of convergence, in view of (9), we must have

1− a1 = 0,a2 = 0,

2a1c22 + 3a2c2

2 − 2a2c3 − a3c22 = 0,

which gives a1 = 1, a2 = 0 and a3 = 2. Therefore, from (9), the error equation of the method (4) becomes

en+1 = (3c32 − c2c3)e4

n +O(

e5n

),

and the assertion follows.

In view of Theorem 1, the proposed fourth order method is


,

xn+1 = xn −(

1 + 2(

f (yn)f (xn)

)2)

f (xn)+ f (yn)f ′(xn)

,(10)

which requires three function evaluations per iteration and consequently is optimal. In addition, theefficiency index of (10) is 1.5874, which is higher than that of (3) having an efficiency index of 1.442.

2.2. Sixth Order Method

Using the results obtained in (10), we propose a new method defined by


,

zn = xn −(

1 + 2(

f (yn)f (xn)

)2)


,

xn+1 = zn − w1(tn)f (zn)f ′(xn)

,

(11)

where w1(tn) = b1 + b2tn is a new weight function and tn is as in (4). The order of convergence isshown in the following result.

Theorem 2. Let f be a real or complex valued function defined in an interval I having a sufficient number ofsmooth derivatives. Let α be a simple root of the equation f (x) = 0 and the initial point x0 is close enough to α.Then, (11) has a sixth order of convergence if b1 = 1 and b2 = 2.

Proof. Let θn = zn − α. Then, from second equation of (11), we obtain

θn = (3c32 − c2c3)e4

n − 2(8c4

2 − 10c22c3 + c2

3 + c2c4)

e5n

+(46c52 − 114c3

2c3 + 30c22c4 + 42c2c2

3 − 3c2c5 − 7c3c4)e6n + O(e7

n).(12)

Now, by expanding f (zn) about α using Equation (12), we obtain

f (zn) = f (θn + α)

= f ′(α)[(3c3

2 − c2c3)

e4n − 2

(8c4

2 − 10c22c3 + c2

3 + c2c4)

e5n

+(46c5

2 − 114c32c3 + 30c2

2c4 + 42c2c23 − 3c2c5 − 7c3c4

)e6

n + O(e7n)].

(13)

Therefore, using (6), (8) and (13) in the third equation of (11), we obtain

en+1 = (1− b1)c2(3c22 − c3)e4

n+(c4

2(22b1 − 3b2 − 16) + c22c3(−22b1 + b2 + 20) + 2(b1 − 1)c2c4 + 2(b1 − 1)c2

3)

e5n

+(c5

2(−90b1 + 31b2 + 46) + c32c3(167b1 − 31b2 − 114) + 2c2

2c4(−17b1 + b2 + 15)+c2(c2

3(−49b1 + 4b2 + 42) + 3(b1 − 1)c5) + 7(b1 − 1)c3c4)e6

n + O(e7n).

(14)

49


In order to obtain sixth order of convergence, the coefficients of e4n and e5

n must vanish in (14), i.e.,b1 = 1 and b2 = 2. Therefore, the error equation of the method (11) becomes

en+1 = c2

(18c4

2 − 9c22c3 + c2

3

)e6

n +O(

e7n

),

and the assertion follows.

In view of Theorem 2, the following is the sixth order method


,

zn = xn −(

1 + 2(

f (yn)f (xn)

)2)


,

xn+1 = zn −(

1 + 2 f (yn)f (xn)

)f (zn)f ′(xn)

.

(15)

2.3. Optimal Eighth Order Method

Notice that the method (15) is not optimal as it requires four function evaluation per iteration toachieve sixth order of convergence. Its efficiency index is 1.5651, which is less than that of the fourthorder method (10). However, an eighth order method is obtained by (10) using an additional Newtonstep. The resulting iterative scheme is


,

zn = xn −(

1 + 2(

f (yn)f (xn)

)2)


,

xn+1 = zn − f (zn)f ′(zn)

.

(16)

Nevertheless, this method requires five function evaluation per iteration, so that its efficiency indexreduces to 1.5157, and, moreover, it is not optimal. Towards making the method (16) more efficientand optimal, we approximate f ′(z) as

f ′(zn) ≈ f ′(xn)

J(tn, un) · G(sn), (17)

where

tn =f (yn)

f (xn), un =

f (zn)

f (xn), sn =

f (zn)

f (yn).

Here, J and G are some appropriate weight functions of two variables and one variable, respectively.This type of approximations was done by Matthies et al. in [22]. Accordingly, we propose thefollowing method:


,

zn = xn −(

1 + 2(

f (yn)f (xn)

)2)


,

xn+1 = zn − f (zn)f ′(xn)

· J(tn, un) · G(sn),

(18)

where tn, un, and sn, are as in (17). For the method (18), we take the functions J and G as

J(tn, un) =1 + 2tn + (β + 2)un + 3t2

n1 + βun

(19)

andG(sn) =

1 + λsn

1 + (λ− 1)sn, (20)

where β and λ belong to C. We prove the following result.

50


Theorem 3. Let f be a real or complex valued function defined on some interval I having a sufficient number ofsmooth derivatives. Let α be a simple root of the equation f (x) = 0 and the initial point x0 is close enough to α.Then, (18) is an eighth order of convergence for the functions J and G given by (19) and (20), respectively.

Proof. In view of (5) and (13), we obtain

un = f (zn)f (xn)

=(3c3

2 − c2c3)

e3n +

(−19c42 + 21c2

2c3 − 2c2c4 − 2c23)

e4n

+(65c5

2 − 138c32c3 + 32c2

2c4 + 45c2c23 − 3c2c5 − 7c3c4

)e5

n + O(e6n).

Similarly, (7) and (13) yield

sn = f (zn)f (yn)

=(3c2

2 − c3)

e2n − 2

(5c3

2 − 6c2c3 + c4)

e3n +

(11c4

2 − 44c22c3 + 17c2c4 + 11c2

3 − 3c5)

e4n

+(56c5

2 + 28c32c3 − 56c2

2c4 − 60c2c23 + 22c2c5 + 30c3c4 − 4c6

)e5

n + O(e6n).

Consequently, (19) gives

J(tn, un) = 1 + 2c2en +(4c3 − 3c2

2)

e2n +

(4c3

2 − 10c2c3 + 6c4)

e3n

+(−3(2β + 1)c4

2 + 2(β + 10)c22c3 − 14c2c4 − 8c2

3 + 8c5)

e4n

+

((47β− 38)c5

2 − (57β + 14)c32c3 + 4(β + 7)c2

2c4

+2c2(4(β + 4)c2

3 − 9c5)− 22c3c4 + 10c6

)e5

n + O(e6n),

(21)

and (20) gives

G(sn) = 1 +(3c2

2 − c3)

e2n − 2

(5c3

2 − 6c2c3 + c4)

e3n

+((20− 9λ)c4

2 + 2(3λ− 25)c22c3 − (λ− 12)c2

3 + 17c2c4 − 3c5)

e4n

+2((30λ− 2)c5

2 + (60− 46λ)c32c3 + 2(3λ− 17)c2

2c4 + c2(6(2λ− 7)c2

3 + 11c5)

+(17− 2λ)c3c4 − 2c6

)e5

n + O(e6n).

(22)

Now, using the values from (6), (12), (13), (21), and (22) in (18), the error equation of the method is

en+1 = c2

(3c2

2 − c3

) (c4

2(6β + 9λ + 9)− 2c22c3(β + 3λ + 4) + c2c4 + c2

3λ)

e8n +O

(e9

n

),

which gives the eighth order of convergence.


In this section, we test the performance of the methods proposed in Section 2 with the help of somenumerical examples. We compare the results obtained with the known methods of the correspondingorder. We consider the following nonlinear equations and initial guesses:

• f1(x) = sin2 x− x2 + 1, x0 = 2,• f2(x) = ln(1 + x2) + exp(x2 − 3x) sin x, x0 = 2,• f3(x) = x2 − (1− x)5, x0 = 1,• f4(x) = x2 − exp(x)− 3x + 2, x0 = 1,• f5(x) =

√x2 + 2x + 5− 2 sin x− x2 + 3, x0 = 2.

In the previous section, we have proved the theoretical order of convergence of various methods.For practical purposes, we can test numerically the order of convergence of these methods by using

51


Approximated Computational Order of Convergence (or ACOC), defined by Cordero and Torregrosa [23].They defined the ACOC of a sequence {xk}, k ≥ 0 as

ACOC =log (|xk+1 − xk| / |xk − xk−1|)

log (|xk − xk−1| / |xk−1 − xk−2|) . (23)

The use of ACOC, given by (23), serves as a practical check on the theoretical error calculations.We apply our proposed methods and other existing methods as discussed in the following

subsections on each of the test functions. Various results of up to four iterations are observed, andwe compare the results obtained at the 4th iteration among different methods of the correspondingorder and shown in Tables 1–3. For a particular test function, we take the same initial guess x0

for each of the methods under consideration. We compare the approximate error Δxn ≡ |xn −xn−1|, the approximate solution xn, the absolute value of corresponding functional value | f (xn)|, andapproximated computational order of convergence (ACOC) at n = 4. In the tables, “NC” stands forno convergence of the method. We use Mathematica 9.0 for the calculations.

3.1. Comparison of the Fourth Order Method

Let us denote our method (10) by M41. We shall compare this method with

• Sharma and Behl method (1), denoted by M42,• Sharifi et al. method (2), denoted by M43,• Jarratt’s method [24], denoted by M44 and given by

yn = xn − 23

f (xn)f ′(xn)

,

xn+1 = xn −(

3 f ′(yn)+ f ′(xn)6 f ′(yn)−2 f ′(xn)

)f (xn)f ′(xn)

,

• Kung–Traub [8] method, denoted by M45, and given by


,

xn+1 = yn −(

f (xn)· f (yn)( f (xn)− f (yn))2

)f (xn)f ′(xn)

.

All the methods M4i, i = 1, 2, 3, 4, 5 are optimal. Table 1 records the performance of allthese methods.

Table 1. Comparison of numerical results of fourth order methods at the 4th iteration.

f1 f2 f3 f4 f5

M41 8.7309 × 10−26 2.7730 × 10−55 9.9454 × 10−30 1.2399 × 10−65 9.2139 × 10−82

M42 1.1188 × 10−27 2.9815 × 10−28 1.0915 × 10−24 7.7434 × 10−72 3.5851 × 10−101

Δxn M43 1.1523 × 10−23 NC 6.1887 × 10−13 1.3049 × 10−15 3.6376 × 10−49

M44 2.0493 × 10−32 2.0594 × 10−31 1.1971 × 10−20 1.5448 × 10−71 1.1488 × 10−97

M45 4.0043 × 10−28 2.8464 × 10−57 2.4018 × 10−30 4.7295 × 10−65 2.8215 × 10−81

M41 1.4045 −7.8835 × 10−218 0.3460 0.2575 2.3320M42 1.4045 −6.9805 × 10−110 0.3460 0.2575 2.3320

xn M43 1.4045 NC 0.3460 0.2575 2.3320M44 1.4045 3.2977 × 10−123 0.3460 0.2575 2.3320M45 1.4045 −3.5010 × 10−226 0.3460 0.2575 2.3320

M41 1.9828 × 10−100 7.8835 × 10−218 1.9230 × 10−116 2.5756 × 10−262 1.1861 × 10−326

M42 4.0436 × 10−108 6.9805 × 10−110 1.1758 × 10−96 6.8107 × 10−287 1.9034 × 10−404

| f (xn)| M43 3.6237 × 10−93 NC 6.4877 × 10−49 7.5782 × 10−62 2.9990 × 10−196

M44 1.7439 × 10−127 3.2977 × 10−123 4.4608 × 10−80 1.3131 × 10−285 2.5652 × 10−390

M45 5.7027 × 10−110 3.5010 × 10−226 9.4841 × 10−120 6.9959 × 10−260 1.1952 × 10−324

M41 3.9919 4.0000 4.0184 4.0000 4.0000M42 3.9935 3.9953 4.0646 4.0000 4.0000

ACOC M43 4.1336 NC 3.5972 4.6265 4.0214M44 3.9978 4.0069 3.9838 4.0000 4.0000M45 3.9946 4.0001 3.9878 4.0000 4.0000

52


3.2. Comparison of Sixth Order Methods

We denote our sixth order method (15) by M61. We shall compare this method with

• M62 : Method of Neta [25] with a = 1, given by


,

zn = yn − f (xn)+a f (yn)f (xn)+(a−2) f (yn)

f (yn)f ′(xn)

,

xn+1 = zn − f (xn)− f (yn)f (xn)−3 f (yn)

f (zn)f ′(xn)

,

• M63 : Method of Grau et al. [26] given by


,

zn = yn − f (yn)f ′(xn)

f (xn)f (xn)−2 f (yn)

,



.

• M64 : Method of Sharma and Guha [27] with a = 2, given by


,



,


f (xn)+a f (yn)f (xn)+(a−2) f (yn)

,

• M65 : Method of Chun and Neta [28] given by


,


1(1− f (yn)

f (xn)

)2 ,

xn+1 = zn − f (yn)f ′(xn)

1(1− f (yn)

f (xn)− f (zn)

f (xn)

)2 .

The comparison of the methods M6i, i = 1, 2, 3, 4, 5 is tabulated in Table 2. From the table, weobserve that the proposed method M61 is compatible with the other existing methods. We can see thatmethod M63 gives different results for the test functions f2 and f5 with given initial guesses.

Table 2. Comparison of numerical results of sixth order methods at the 4th iteration.

f1 f2 f3 f4 f5

M61 1.8933 × 10−73 1.8896 × 10−148 5.1627 × 10−90 1.3377 × 10−199 9.5891 × 10−261

M62 1.6801 × 10−106 2.9382 × 10−152 2.4137 × 10−64 1.7893 × 10−191 3.75383 × 10−255

Δxn M63 2.9803 × 10−95 2.9803 × 10−95 2.9815 × 10−82 2.9815 × 10−82 2.9803 × 10−95

M64 5.0012 × 10−85 2.4246 × 10−153 4.9788 × 10−69 4.6397 × 10−198 4.0268 × 10−259

M65 9.9516 × 10−88 2.1737 × 10−154 3.3993 × 10−86 2.7764 × 10−193 3.4903 × 10−256

M61 1.4045 −1.1331 × 10−884 0.3460 0.2575 2.3320M62 1.4045 4.5753 × 10−908 0.3460 0.2575 2.3320

xn M63 1.4045 1.4045 0.3460 0.2575 1.4045M64 1.4045 1.0114 × 10−914 0.3460 0.2575 2.3320M65 1.4045 −3.7511 × 10−921 0.3460 0.2575 2.3320

53


Table 2. Cont.

f1 f2 f3 f4 f5

M61 5.6523 × 10−436 1.1331 × 10−884 1.8046 × 10−535 0.0 0.0M62 6.7308 × 10−636 4.5753 × 10−908 1.0347 × 10−381 0.0 0.0

| f (xn)| M63 8.1802 × 10−568 8.1802 × 10−568 8.2004 × 10−490 8.2004 × 10−490 8.1802 × 10−568

M64 5.7605 × 10−506 1.0114 × 10−914 1.8726 × 10−409 0.0 0.0M65 3.7794 × 10−522 3.7511 × 10−921 4.8072 × 10−514 0.0 0.0

M61 5.9980 6.0000 5.9980 6.0000 6.0000M62 5.9992 6.0000 5.9854 6.0000 6.000

ACOC M63 5.9997 5.9997 5.9992 5.9992 5.9997M64 5.9991 6.0000 5.9984 6.0000 6.0000M65 5.9993 6.0000 6.0088 6.0000 6.0000

3.3. Comparison of Eighth Order Methods

Consider the eighth order method (18), which involves the parameter pair (β, λ). We denote

• M81 the case where (β, λ) = (0, 0), whose iterative expression results in


,

zn = xn − f (xn)+ f (yn)f ′(xn)

(1 + 2

(f (yn)f (xn)

)2)

,


(1+2tn+2un+3t2

n1−sn

),

• M82 for (β, λ) = (1, 1), resulting in the iterative scheme given by M81 :


,


(1 + 2

(f (yn)f (xn)

)2)

,


(1+2tn+3un+3t2

n1+un

(1 + sn))

,

• M83 for (β, λ) = (0, 1), whose iterative method is


,


(1 + 2

(f (yn)f (xn)

)2)

,


((1 + 2tn + 2un + 3t2

n)(1 + sn))

.

Along with these, we take the following methods for the comparison of numerical results:

• Matthies et al. in [22] presented an optimal class of 8th order method from the Kung–Traubmethod [8]. For some particular values of the parameters, one of the methods denoted by M84 isgiven by


,

zn = yn −(

f (xn) f (yn)( f (xn)− f (yn))2

)f (xn)f ′(xn)

,


(2+tn+5un+4t2

n+4t3n

2−3tn+un+2t2n· 2+sn

2−sn

).

54


• Babajee et al. in [11] presented a family of eighth order methods. For some fixed values ofparameters, the method denoted by M85 is given by


(1 + ( f (xn)

f ′(xn))5)

,


(1− f (yn)

f (xn)

)−2,


((1+t2

n+5t4n+sn

(1−tn−un)2

).

• Chun and Lee in [29] presented a family of optimal eighth order methods. For some particularvalues of parameters, the method denoted by M86 is given by


,


1(1− f (yn)

f (xn)

)2 ,


1(1−tn− t2n

2 +t3n2 − un

2 − sn2

)2 .

In all the above methods, tn, un and sn are as given in (17). The performance of the methods M8i,i = 1, 2, 3, 4, 5, 6 are recorded in Table 3.

Table 3. Comparison of numerical results of eighth order methods at the 4th iteration.

f1 f2 f3 f4 f5

Δxn

M81 5.8768 × 10−187 1.5404 × 10−393 2.5345 × 10−165 6.1099 × 10−495 4.4344 × 10−658

M82 2.0563 × 10−165 9.0158 × 10−321 1.1101 × 10−167 5.4494 × 10−421 4.0437 × 10−598

M83 4.5429 × 10−170 1.5139 × 10−324 2.9710 × 10−168 2.8838 × 10−421 2.9107 × 10−604

M84 2.4469 × 10−187 4.9438 × 10−351 4.3825 × 10−171 1.8592 × 10−438 4.3404 × 10−614

M85 2.6744 × 10−204 NC 1.7766 × 10−177 6.5231 × 10−192 9.8976 × 10−553

M86 4.1482 × 10−235 1.3271 × 10−380 5.6991 × 10−175 2.5934 × 10−455 7.1011 × 10−617

xn

M81 1.4045 0.0 0.3460 0.2575 2.3320M82 1.4045 0.0 0.3460 0.2575 2.3320M83 1.4045 0.0 0.3460 0.2575 2.3320M84 1.4045 0.0 0.3460 0.2575 2.3320M85 1.4045 NC 0.3460 0.2575 2.3320M86 1.4045 0.0 0.3460 0.2575 2.3320

| f (xn)|

M81 0.0 0.0 0.0 0.0 0.0M82 0.0 0.0 0.0 0.0 0.0M83 0.0 0.0 0.0 0.0 0.0M84 0.0 0.0 0.0 0.0 0.0M85 0.0 0.0 0.0 0.0 0.0M86 0.0 0.0 0.0 0.0 0.0

ACOC

M81 7.9999 8.0000 7.9993 8.0000 8.0000M82 7.9996 8.0000 8.0000 8.0000 8.0000M83 7.9997 8.0000 7.9996 8.0000 8.0000M84 7.9998 8.0000 8.0047 8.0000 8.0000M85 7.9995 NC 8.0020 8.0004 8.0000M86 8.0000 8.0000 8.0023 8.0000 8.0000

From Tables 1–3, we observe that the proposed methods are compatible with other existingmethods (and sometimes perform better than other methods) of the corresponding order. Not anyparticular method is superior to others for all examples. Among the family of eighth order methods (18),from Table 3, we observe that the method M81 performs better than other two. For more understandingabout the iterative methods, we study the dynamics of these methods in the next section.

55


4. Applications

The applications discussed in Sections 4.1–4.3 are based on standard engineering examples, andwe refer to [30]. We use the proposed methods M41, M61, and M8i, i = 1, 2, 3 to obtain the variousresults from the first three iterations of these examples. In particular, we compute the value of theunknowns xn−1 and xn, absolute value of the function f (xn) and absolute value of the difference d ofunknown in two consecutive iterations, i.e., d = |xn − xn−1|, n = 1, 2, 3.

4.1. Pipe Friction Problem

Determining fluid flow through pipes and tubes has great relevance in many areas of engineeringand science. In engineering, typical applications include the flow of liquids and gases throughpipelines and cooling systems. Scientists are interested in topics ranging from flow in blood vesselsto nutrient transmission through a plant’s vascular system. The resistance to flow in such conduitsis parameterized by a dimensionless number called the friction factor f . For a flow with turbulence,the Colebrook equation [31] provides a means to calculate the friction factor:

0 =1√

f+ 2.0 log

(ε

3.7D+

2.51Re√

f

), (24)

where ε is the roughness (m), D is the diameter (m) and Re is the Reynolds number

Re =ρvD

μ.

Here, ρ denotes the fluid density (kg/m3), v the velocity of the fluid (m/s) and μ the dynamicalviscosity (N·s/m2). A flow is said to be turbulent if Re > 4000.

To determine f for air flow through a smooth and thin tube, the parameters are taken to beρ = 1.23 kg/m3, μ = 1.79× 10−5 N·s/m2, D = 0.005 m, V = 40 m/s and ε = 0.0000015 m. Since thefriction factors range from about 0.008 to 0.08, we choose initial guess f0 = 0.023. To determine theapproximate value of f , we use the function

g( f ) =1√

f+ 2.0 log

(ε

3.7D+

2.51Re√

f

). (25)

The results obtained by the various methods are presented in Table 4.

Table 4. Results of pipe friction problem.

# Iter Value M41 M61 M81 M82 M83

f 0.0169 0.0170 0.0170 0.0170 0.01701 g( f ) 0.0240 0.0104 0.0009 0.0005 0.0008

d 0.0061 0.0060 0.0060 0.0060 0.0060

f 0.0170 0.0170 0.0170 0.0170 0.01702 g( f ) 3.0954 × 10−9 2.6645 × 10−15 8.8818 × 10−16 8.8818 × 10−16 8.8818 × 10−16

d 0.0001 4.1700 × 10−5 3.7223 × 10−6 2.0962 × 10−6 3.3172 × 10−6

f 0.0170 0.0170 0.0170 0.0170 0.01703 g( f ) 8.8818 × 10−16 8.8818 × 10−16 8.8818 × 10−16 8.8818 × 10−16 8.8818 × 10−16

d 1.2442 × 10−11 1.0408 × 10−17 6.9389 × 10−18 0.0 0.0

4.2. Open-Channel Flow

An open problem in civil engineering is to relate the flow of water with other factors affecting theflow in open channels such as rivers or canals. The flow rate is determined as the volume of water

56


passing a particular point in a channel per unit time. A further concern is related to what happenswhen the channel is slopping.

Under uniform flow conditions, the flow of water in an open channel is given byManning’s equation

Q =

√S

nAR2/3, (26)

where S is the slope of the channel, A is the cross-sectional area of the channel, R is the hydraulicradius of the channel and n is the Manning’s roughness coefficient. For a rectangular channel havingthe width B and the defth of water in the channel y, it is known that

A = By

andR =

ByB + 2y

.

With these values, (26) becomes

Q =

√S

nBy(

ByB + 2y

)2/3. (27)

Now, if it is required to determine the depth of water in the channel for a given quantity of water, (27)can be rearranged as

f (y) =√

Sn

By(

ByB + 2y

)2/3−Q. (28)

In our work, we estimate y when the remaining parameters are assumed to be given as Q = 14.15 m3/s,B = 4.572 m, n = 0.017 and S = 0.0015. We choose as an initial guess y0 = 4.5 m. The results obtainedby the various methods are presented in Table 5.

Table 5. Results of an open channel problem.


y 1.4804 1.4666 1.4652 1.4653 1.46531 f (y) 0.2088 0.0204 0.0016 0.0029 0.0028

d 3.0200 3.0334 3.0348 3.0347 3.0347

y 1.4651 1.4651 1.4651 1.4651 1.46512 f (y) 4.5027 × 10−9 1.7764 × 10−15 × 10−15 3.5527 × 10−15 3.5527 × 10−15

d 0.0154 0.0015 0.0001 0.0002 0.0002

y 1.4651 1.4651 1.4651 1.4651 1.46513 f (y) 3.5527 × 10−15 7.1054 × 10−14 6.5725 × 10−14 5.3291 × 10−15 1.7764 × 10−15

d 3.3152 × 10−10 5.1070 × 10−15 5.1070 × 10−15 6.6613 × 10−16 2.2204 × 10−16

4.3. Ideal and Non-Ideal Gas Laws

The ideal gas law isPV = nRT,

where P is the absolute pressure, V is the volume, n is the number of moles, R is the universal gasconstant and T is the absolute temperature. Due to its limited use in engineering, an alternativeequation of state for gases is the given van der Waals equation [32–35](

P +a

v2

)(v− b) = RT,

57


where v = Vn is the molal volume and a, b are empirical constants that depend on the particular gas.

The computation of the molal volume is done by solving

f (v) =(

P +a

v2

)(v− b)− RT. (29)

We take the remaining parameters as R = 0.082054 L atm/(mol K), for carbon dioxide a = 3.592,b = 0.04267, T = 300 K, p = 1 atm, and the initial guess for the molal volume is taken as v0 = 3.The results obtained by the various methods are presented in Table 6. In this table, IND stands forindeterminate form.

Table 6. Numerical results of ideal and non-ideal gas law.


v 26.4881 27.0049 23.9583 24.1631 24.02741 f (v) 1.9647 2.4788 0.5509 0.3474 0.4823

d 23.4881 24.0049 20.9583 21.1631 21.0274

v 24.5126 24.5126 24.5126 24.5126 24.51262 f (v) 2.7340 × 10−8 3.3573 × 10−12 0.0 0.0 0.0

d 1.9756 2.4923 0.5543 0.3495 0.4852

v 24.5126 24.5126 IND IND IND3 f (v) 0.0 0.0 IND IND IND

d 2.7503 × 10−8 3.3786 × 10−12 IND IND IND

5. Dynamical Analysis

The stability analysis of the methods M41, M61 and M8i, i = 1, 2, 3, is performed in this section.The dynamics of the proposed methods on a generic quadratic polynomial will be studied, analyzingthe associated rational operator for each method. This analysis shows their performance dependingon the initial estimations. In addition, method M41 is analyzed for cubic polynomials. First, we recallsome basics on complex dynamics.

5.1. Basics on Complex Dynamics

Let R : C −→ C be a rational function defined on the Riemann sphere. Let us recall that everyholomorphic function from the Riemann sphere to itself is in fact a rational function R(z) = P(z)

Q(z) ,where P and Q are complex polynomials (see [36]). For older work on dynamics on the Riemannsphere, see, e.g., [37].

The orbit of a point z0 ∈ C is composed by the set of its images by R, i.e.,

{z0, R(z0), R2(z0), . . . , Rn(z0), . . .}.

A point zF ∈ C is a fixed point if R(zF) = zF. Note that the roots z∗ of an equation f (z) = 0 are fixedpoints of the associated operator of the iterative method. Fixed points that do not agree with a root off (x) = 0 are strange fixed points.

The asymptotical behavior of a fixed point zF is determined by the value of its multiplierμ = |R′(zF)|. Then, zF is attracting, repelling or neutral if μ is lower, greater or equal to 1, respectively.In addition, it is superattracting when μ = 0.

For an attracting fixed point zF, its basin of attraction is defined as the set of its pre-images ofany order:

A(zF) = {z0 ∈ C : Rn(z0) −→ zF, n → ∞}.

The dynamical plane represents the basins of attraction of a method. By iterating a set of initialguesses, their convergence is analyzed and represented. The points zC ∈ C that satisfy R′(zC) = 0 arecalled critical points of R. When a critical point does not agree with a solution of f (x) = 0, it is a free

58


critical point. A classical result [21] establishes that there is at least one critical point associated witheach immediate invariant Fatou component.

5.2. Rational Operators

Let p(z) be a polynomial defined on C. Corresponding to the methods developed in this paper,i.e., methods (10), (15) and family (18), we define the operators R4(z), R6(z) and R8(z), respectively,in C as follows:

R4(z) = z−(

1 + 2(

p(y(z))p(z)

)2)

p(z) + p(y(z))p′(z) , (30)

R6(z) = R4(z)−(

1 + 2p(y(z))

p(z)

)p(R4(z))

p′(z) ,

R8(z) = R4(z)− p(R4(z))p′(z) J(z)G(z),

where y(z) = z− p(z)p′(z) and

J(z) =1+2 p(y(z))

p(z) +(β+2) p(R4(z))p(z) +3

(p(y(z))

p(z)

)2

1+βp(R4(z))

p(z)

,

G(z) =1+λ

p(R4(z))p(y(z))

1+(λ−1) p(R4(z))p(y(z))

.

First, we recall the following result for the generalization of the dynamics of M41.

Theorem 4 (Scaling Theorem for method M41). Let f (z) be an analytic function in the Riemann sphere andlet A(z) = ηz + σ, with η �= 0, be an affine map. Let h(z) = μ( f ◦ A)(z) with μ �= 0. Then, the fixed pointoperator R f

4 is affine conjugated to Rh4 by A, i.e.,

(A ◦ Rh4 ◦ A−1)(z) = R f

4(z).

Proof. From (30), let the fixed point operators associated with f and h be, respectively,

R f4(z) = z−

(1 + 2

(f (y(z))

f (z)

)2)

f (z)+ f (y(z))f ′(z) ,

Rh4(z) = z−

(1 + 2

(h(y(z))

h(z)

)2)

h(z)+h(y(z))h′(z) .

Thus, we have

(R f4 ◦ A)(z) = A(z)−

(1 + 2

f 2(A(y))f 2(A(z))

)f (A(z)) + f (A(y))

f ′(A(z)). (31)

Being h′(z) = ημ f ′(A(z)), we obtain

Rh4(z) = z−

(1 + 2 μ2 f 2(A(y))

μ2 f 2(A(z))

)μ f (A(z))+μ f (A(y))

ημ f ′(A(z))

= z−(

1 + 2 f 2(A(y))f 2(A(z))

)f (A(z))+ f (A(y))

η f ′(A(z)) .

59


The affine map A satisfies A(z1 − z2) = A(z1)− A(z2) + σ, ∀z1, z2. Then, from (32), we have

(A ◦ Rh4)(z) = A(z)− A

((1 + 2 f 2(A(y))

f 2(A(z))

)f (A(z))+ f (A(y))

η f ′(A(z))

)+ σ

= A(z)−(

η(

1 + 2 f 2(A(y))f 2(A(z))

)f (A(z))+ f (A(y))

η f ′(A(z)) + σ)+ σ

= A(z)−(

1 + 2 f 2(A(y))f 2(A(z))

)f (A(z))+ f (A(y))

f ′(A(z)) .

Thus, it proves that (R f4 ◦ A)(z) = (A ◦ Rh

4)(z) and then method M41 satisfies the Scaling Theorem.

Theorem 4 allows for generalizing the dynamical study of a specific polynomial to a genericfamily of polynomials by using an affine map. Analogous to the way we proved the Scaling Theoremfor the operator R4, it also follows that the fixed point operators R6 and R8 obey the Scaling Theorem.

5.3. Dynamics on Quadratic Polynomials

The application of the rational functions on a generic quadratic polynomial p(z) = (z− a)(z− b),a, b ∈ C is studied below. Let R4,a,b be the rational operator associated with method M41 on p(z).When the Möbius transformation h(u) = a−u

b−u is applied to R4,a,b, we obtain

S4(z) = (h ◦ R4,a,b ◦ h−1)(z) =z4 (z4 + 6z3 + 14z2 + 14z + 3

)3z4 + 14z3 + 14z2 + 6z + 1

. (32)

The rational operator associated with M41 on p(z) does not depend on a and b. Then, the dynamicalanalysis of the method on all quadratic polynomials can be studied through the analysis of (32).In addition, the Möbius transformation h maps its roots a and b to z∗1 = 0 and z∗2 = ∞, respectively.

The fixed point operator S4(z) has nine fixed points: zF1 = 0 and zF

2 = ∞, which are superattracting,

and zF3 = 1, zF

4,5 = 12 (−3±√5), zF

6−7 = −2+√

22 ± i

√32 −

√2, zF

8−9 = −2−√22 ± i

√32 +

√2, all of them

being repelling. Computing S′4(z) = 0, 5 critical points can be found. zC1,2 = z∗1,2 and the free critical

points zC3 = −1 and zC

4,5 = 16 (−13±√133).

Following the same procedure, when Möbius transformation is applied to methods M6 and M8i,i = 1, 2, 3, on polynomial p(z), the respective fixed point operators turn into

S6(z) =z6(z12+16z11+119z10+544z9+1700z8+3808z7+6206z6+7288z5+5973z4+3248z3+1111z2+216z+18)

18z12+216z11+1111z10+3248z9+5973z8+7288z7+6206z6+3808z5+1700z4+544z3+119z2+16z+1 ,

S81(z) =P30(z)P22(z)

, S82(z) =P42(z)P34(z)

, S83(z) =Q42(z)Q34(z)

,

where Pk and Qk denote polynomials of degree k.The fixed point operator S6 has 19 fixed points: the two superattracting fixed points zF

1,2 = z∗1,2,the repelling fixed point zF

3 = 1 and the repelling fixed points zF4 , . . . , zF

19, which are the roots of asixteenth-degree polynomial.

Regarding the critical points of S6, the roots of p(z) are critical points, and S6 has the free criticalpoints zC

3 = −1 and the roots of a tenth-degree polynomial, zC4 , . . . , zC

11.The dynamical planes are a useful tool in order to analyze the stability of an iterative method.

Taking each point of the plane as initial estimation to start the iterative process, they represent theconvergence of the method depending on the initial guess. In this sense, the dynamical planes showthe basins of attraction of the attracting points.

Figure 1 represents the dynamical planes of the methods S4 and S6. The generation of thedynamical planes follows the guidelines established in [38]. A mesh of 500× 500 complex values hasbeen set as initial guesses in the intervals −5 < �{z} < 5, −5 < �{z} < 5. The roots z∗1 = 0 andz∗2 = ∞ are mapped with orange and blue colors, respectively. The regions where the colors are darkerrepresent that more iterations are necessary to converge than with the lighter colors, with a maximum

60


of 40 iterations of the methods and a stopping criteria of a difference between two consecutive iterationslower than 10−6.

As Figure 1 illustrates, there is convergence to the roots for every initial guess. Let us remark that,when the order of the method increases, the basin of attraction of z∗1 = 0 becomes more intricate.

Finally, for the fixed point operators associated with family M8, the solutions of S8i(z) = zfor i = 1, 2, 3 give the superattracting fixed points zF

1,2 = z∗1,2 and the repelling point zF3 = 1. In

addition, S81 has 28 repelling points. S82 and S83 have 38 repelling points, corresponding to the rootsof polynomials of 28 and 38 degree, respectively, and the strange fixed points zF

4,5 = 12 (−1±√5).

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(a) S4

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(b) S6

Figure 1. Dynamical planes of methods S4 and S6.

The number of critical points of the fixed point operators S8i are collected in Table 7. In addition,the number of strange fixed points and free critical points are also included in the table for all ofthe methods.

Table 7. Number of strange fixed points (SFP) and free critical points (FCP) for the methods onquadratic polynomials.

S4 S6 S81 S82 S83

Strange fixed points 7 17 29 41 41Free critical points 3 29 29 43 29

Figure 2 represents the dynamical planes of the methods S81, S82 and S83. Since the originalmethods satisfy the Scaling Theorem, the generation of one dynamical plane involves the study ofevery quadratic polynomial.

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(a) S81

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(b) S82

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(c) S83

Figure 2. Dynamical planes of methods S8i, i = 1, 2, 3.

61


There is an intricate region around z = −1 in Figure 2a, becoming wider in Figure 2b,c aroundz = −1.5. However, for every initial guess in the three dynamical planes of Figure 2, there isconvergence to the roots.

5.4. Dynamics on Cubic Polynomials

The stability of method M41 on cubic polynomials is analyzed below. As stated by the authorsin [39], the Scaling Theorem reduces the dynamical analysis on cubic polynomials to the study ofdynamics on the cubic polynomials p0(z) = z3, p+(z) = z3 + z, p−(z) = z3 − z and the family ofpolynomials pγ(z) = z3 + γz + 1. Let us recall that the first one only has the root z∗1 = 0, while p+(z)and p−(z) have three simple roots: z∗1 = 0 and z∗2,3 = ∓i or z∗2,3 = ∓1, respectively. For each γ ∈ C,the polynomial pγ(z) also has three simple roots that depend on the value of γ. They will be denotedby z∗1,2,3(γ).

By applying method M41 to polynomials p0(z), p+(z) and p−(z), the fixed point operatorsobtained are, respectively,

S4,0(z) = 46z81 , S4,+(z) = 6z5+36z7+46z9

(1+3z2)4 , S4,−(z) = 6z5−36z7+46z9

(1−3z2)4 .

The only fixed point of S4,0(z) agrees with the root of the polynomial, so it is superattracting, and theoperator does not have critical points.

The rest of the fixed point operators have six repelling fixed points, in addition to the roots of the

corresponding polynomials: zF4,5 = ± i

√5

5 and zF6−9 = ±i

√17 (3±

√2) for S4,+(z), and zF

4,5 = ±√

55 and

zF6−9 = ±

√17 (3±

√2) for S4,−(z).

Regarding the critical points of S4,+(z) and S4,−(z), they match with the roots of the polynomials.

Moreover, there is the presence of free critical points with values zC4,5 = ±i

√523 for S4,+(z) and

zC4,5 = ±

√5

23 for S4,−(z).As for quadratic polynomials, the dynamical planes of method M41 when it is applied to the

cubic polynomials have been represented in Figure 3. Depending on the roots of each polynomial, theconvergence to z∗1 = 0 is represented in orange, while the convergence to z∗2 and z∗3 is represented inblue and green, respectively. It can be see in Figure 3 that there is full convergence to a root in the threecases. However, there are regions with darker colors that indicate a higher number of iterations untilthe convergence is achieved.

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(a) p0(z)

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(b) p+(z)

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(c) p−(z)

Figure 3. Dynamical planes of method M41 on polynomials p0(z), p+(z) and p−(z).

When method M41 is applied on pγ(z), the fixed point function turns into

S4,γ(z) = −γ3 − 46z9 − 36γz7 + 42z6 − 6γ2z5 + 45γz4 + 6z3 + 12γ2z2 − 1

(γ + 3z2)4 .

62


The fixed points of S4,γ(z) are the roots of the polynomial z∗1,2,3(γ), being superattracting, and thestrange fixed points zF

4−9(γ) that are the roots of the sixth-degree polynomial q(z, γ) = 35z6 + 37γz4 +

7z3 + 11γ2z2 + γz + γ3 − 1.As the asymptotical behavior of zF

4 (γ), . . . , zF9 (γ) depends on the value of γ, the stability planes

corresponding to these points are represented in Figure 4. For each strange fixed point, a meshof 100 × 100 points covers the values of �(γ) ∈ [−5, 5] and �(γ) ∈ [−5, 5]. The stability planeshows the values for the parameter where |S′4,γ(z

F)| is lower or greater than 1, represented in red orgreen, respectively.

-5 0 5{ }

-5

0

5

{}

Figure 4. Stability planes of zF4−9(γ).

From Figure 4, the strange fixed points are always repelling for (�(γ),�(γ)) ∈ [−5, 5]× [−5, 5].Then, the only attracting fixed points are the roots of the polynomial. This fact guarantees a betterstability of the method.

The solutions of S′4,γ(z) = 0 are the critical points zC1,2,3(γ) = z∗1,2,3(γ) and the free critical points

zC4 = 0 and

zC5 (γ) =

(√69√

125γ3+2484+414)2/3−5 3√69γ

692/3 3√√

69√

125γ3+2484+414,

zC6,7(γ) =

(−1±i√

3)(√

69√

125γ3+2484+414)2/3

+5 3√69(1±i√

3)γ

2 692/3 3√√

69√

125γ3+2484+414.

When the fixed point function has dependence on a parameter, another useful representation is theparameters’ plane. This plot is generated in a similar way to the dynamical planes, but, in this case,by iterating the method taking as an initial guess a free critical point and varying the value of γ in acomplex mesh of values, so each point in the plane represents a method of the family. The parameters’plane helps to select the values for the parameter that give rise to the methods of the family withmore stability.

The parameters’ planes of the four free critical points are shown in Figure 5. Parameter γ takes thevalues of 500× 500 points in a complex mesh in the square [−5, 5]× [−5, 5]. Each point is representedin orange, green or blue when the corresponding method converges to an attracting fixed point. Theiterative process ends when the maximum number of 40 iterations is reached, in which case the pointis represented in black, or when the method converges as soon as, by the stopping criteria, a differencebetween two consecutive iterations lower than 10−6 is reached.

For the parameters’ planes in Figure 5, there is not any black region. This guarantees that thecorresponding iterative schemes converge to a root of pγ(z) for all the values of γ.

In order to visualize the basins of attraction of the fixed points, several values of γ have beenchosen to perform the dynamical planes of method M41. These values have been selected from thedifferent regions of convergence observed in the parameters planes. Figure 6, following the same codeof colours and stopping criteria as in the other representations, shows the dynamical planes obtainedwhen these values of γ are fixed.

63


-5 0 5

{ }

-5

-4

-3

-2

-1

0

1

2

3

4

5

{}

(a) zC4

-5 0 5

{ }

-5

-4

-3

-2

-1

0

1

2

3

4

5

{}

(b) zC5 (γ)

-5 0 5

{ }

-5

-4

-3

-2

-1

0

1

2

3

4

5

{}

(c) zC6 (γ)

-5 0 5

{ }

-5

-4

-3

-2

-1

0

1

2

3

4

5

{}

(d) zC7 (γ)

Figure 5. Parameter planes of the critical points of method M41 on pγ(z).

As Figure 6 shows, there is not any initial guess that tends to a point different than the roots. Thisfact guarantees the stability of these methods on the specific case of any cubic polynomial.

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(a) γ = −2 + 4i

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(b) γ = −1 + i

-5 0 5

{z}

-5

-4

-3

-2

-1

0

1

2

3

4

5

{z}

(c) γ = 0.5− 0.5i

Figure 6. Dynamical planes for method M41 on pγ(z) for different values of γ.

6. Conclusions

Two iterative schemes of orders of convergence four and six, and a family of methods of ordereight have been introduced. The method of order four and the family of order eight are optimalin the sense of Kung–Traub’s conjecture. The development of the order of convergence of everymethod has been performed. For every method, we have made a numerical experiment, over both testfunctions and real engineering problems. In order to analyze the stability of the introduced methods,the dynamical behavior of them has been studied. The results confirm that the methods have widebasins of attraction, guaranteeing the stability over some nonlinear problems.

Author Contributions: The individual contributions of the authors are as follows: conceptualization, P.J.;validation, P.B.C. and P.J.; formal analysis, F.I.C.; writing, original draft preparation, N.G. and F.I.C.; numericalexperiments, N.G. and P.B.C.


Acknowledgments: The second and third authors have been partially supported by PGC2018-095896-B-C22(MCIU/AEI/FEDER/UE) and Generalitat Valenciana PROMETEO/2016/089.


References

1. Ostrowski, A.M. Solution of Equations and Systems of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.2. Cordero, A.; Fardi, M.; Ghasemi, M.; Torregrosa, J.R. Accelerated iterative methods for finding solutions of

nonlinear equations and their dynamical behavior. Calcolo 2012, 51, 17–30. [CrossRef]3. Lotfi, T.; Sharifi, S.; Salimi, M.; Siegmund, S. A new class of three-point methods with optimal convergence

order eight and its dynamics. Numer. Algor. 2015, 68, 261–288. [CrossRef]4. Chicharro, F.I.; Cordero, A.; Garrido, N.; Torregrosa, J.R. Wide stability in a new family of optimal

fourth-order iterative methods. Comput. Math. Meth. 2019, 1, e1023. [CrossRef]

64


5. Chicharro, F.I.; Cordero, A.; Torregrosa, J.R. Dynamics of iterative families with memory based on weightfunctions procedure. J. Comput. Appl. Math. 2019, 354, 286–298. [CrossRef]

6. Sharma, R.; Bahl, A. An optimal fourth order iterative method for solving nonlinear equations and itsdynamics. J. Complex Anal. 2015, 2015, 259167. [CrossRef]

7. Sharifi, M.; Babajee, D.K.R.; Soleymani, F. Finding solutions of nonlinear equations by a class of optimalmethods. Comput. Math. Appl. 2012, 63, 764–774. [CrossRef]

8. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iterations. J. Assoc. Comput. Mach. 1974,21, 643–651. [CrossRef]

9. Potra, F.A.; Pták, V. Nondiscrete induction and iterative processes. Res. Notes Math. 1984, 103, 112–119.10. Amat, S.; Busquier, S.; Plaza, S. Review of some iterative root finding methods from a dynamical point of

view. Scientia 2004, 10, 3–35.11. Babajee, D.K.R.; Cordero, A.; Soleymani, F.; Torregrosa, J.R. On improved three-step schemes with high

efficiency index and their dynamics. Numer. Algor. 2014, 65, 153–169. [CrossRef]12. Chicharro, F.I.; Cordero, A.; Torregrosa, J.R.; Vassileva, M.P. King-type derivative-free iterative families: Real

and memory dynamics. Complexity 2017, 2017, 2713145. [CrossRef]13. Cordero, A.; Feng, L.; Magreñán, Á.A.; Torregrosa, J.R. A new fourth order family for solving nonlinear

problems and its dynamics. J. Math. Chem. 2015, 53, 893–910. [CrossRef]14. Scott, M.; Neta, B.; Chun, C. Basin attractors for various methods. Appl. Math. Comput. 2011, 218, 2584–2599.

[CrossRef]15. Varona, J.L. Graphic and numerical comparison between iterative methods. Math. Intell. 2002, 24, 37–46.

[CrossRef]16. Vrscay, E.R.; Gilbert, W.J. Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and

Konig rational iteration functions. Numer. Math. 1988, 52, 1–16. [CrossRef]17. Alexander, D.S. A History of Complex Dynamics: From Schröder to Fatou and Julia; Vieweg & Teubner Verlag:

Wiesbaden, Germany, 1994.18. Beardon, A.F. Iteration of Rational Functions; Springer: New York, NY, USA, 2000.19. Blanchard, P. The dynamics of Newton’s method. Proc. Sympos. Appl. Math. 1994, 49, 139–154.20. Carleson, L.; Gamelin, T.W. Complex Dynamics; Springer-Verlag: New York, NY, USA, 1992.21. Devaney, R.L. An Introduction to Chaotic Dynamical Systems; Addison-Wesley: Redwood City, CA, USA, 1989.22. Matthies, G.; Salimi, M.; Sharifi, S.; Varona, J.L. An optimal class of eighth-order iterative methods based on

Kung and Traub’s method with its dynamics. arXiv 2015, arXiv:1508.01748v1.23. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth order quadrature formulas. Appl. Math.

Comput. 2007, 190, 686–698. [CrossRef]24. Jarratt, P. Some efficient fourth order multipoint methods for solving equations. BIT 1969, 9, 119–124.

[CrossRef]25. Neta, B. A sixth-order family of methods for nonlinear equations. Int. J. Comput. Math. 1979, 7, 157–161.

[CrossRef]26. Grau, M.; Díaz-Barrero, J.L. An improvement to Ostrowski root-finding method. Appl. Math. Comput. 2006,

173, 450–456. [CrossRef]27. Sharma, J.R.; Guha, R.K. A family of modified Ostrowski’s methods with accelerated sixth order convergence.

Appl. Math. Comput. 2007, 190, 111–115.28. Chun, C.; Neta, B. A new sixth-order scheme for nonlinear equations. Appl. Math. Lett. 2012, 25, 185–189.

[CrossRef]29. Chun, C.; Lee, M.Y. A new optimal eighth-order family of iterative methods for the solution of nonlinear

equations. Appl. Math. Comput. 2013, 223, 506–519. [CrossRef]30. Chapra, S.C.; Canale, R.C. Numerical Methods for Engineers; McGraw Hills Education: New York, NY, USA,

2015.31. Brkic, D. A note on explicit approximations to Colebrook’s friction factor in rough pipes under highly

turbulent cases. Int. J. Heat Mass tramsf. 2016, 93, 513–515. [CrossRef]32. Wang, J.; Pang, Y.; Zhang, Y. Limits of solutions to the isentropic Euler equations for van der Waals gas.

Int. J. Nonlinear Sci. Numer. Simul. 2019, 20, 461–473. [CrossRef]33. Gates, D.J.; Penrose, O. The van der Waals limit for classical systems. I. A variational principle. Comm. Math.

Phys. 1969, 15, 255–276. [CrossRef]

65


34. Gates, D.J.; Penrose, O. The van der Waals limit for classical systems. II. Existence and continuity of thecanonical pressure. Comm. Math. Phys. 1970, 16, 231–237. [CrossRef]

35. Gates, D.J.; Penrose, O. The van der Waals limit for classical systems. III. Deviation from the van derWaals-Maxwell theory. Comm. Math. Phys. 1970, 17, 194–209. [CrossRef]

36. Blanchard, P. Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 1984, 11, 85–141.[CrossRef]

37. Schlag, W. A Course in Complex Analysis and Riemann Surfaces; American Mathematical Society: Providence,RI, USA, 2014.

38. Chicharro, F.I.; Cordero, A.; Torregrosa, J.R. Drawing dynamical and parameters planes of iterative familiesand methods. Sci. World J. 2013, 2013, 780153. [CrossRef] [PubMed]

39. Amat, S.; Busquier, S.; Plaza, S. Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl.2010, 366, 24–32. [CrossRef]


66

mathematics

Article

Higher-Order Derivative-Free Iterative Methods forSolving Nonlinear Equations and Their Basinsof Attraction

Jian Li 1,*, Xiaomeng Wang 1 and Kalyanasundaram Madhu 2,*

1 Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China;[email protected]

2 Department of Mathematics, Saveetha Engineering College, Chennai 602105, India* Correspondence: [email protected] (J.L.); [email protected] (K.M.)

Received: 08 September 2019; Accepted: 16 October 2019; Published: 4 November 2019

Abstract: Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-orderalgorithms for solving one-variable equations. The new methods are fourth-, eighth-,and sixteenth-order converging and require at each iteration three, four, and five function evaluations,respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture;the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have givenconvergence analyses of the proposed methods and also given comparisons with already establishedknown schemes having the same convergence order, demonstrating the efficiency of the presenttechniques numerically. We also studied basins of attraction to demonstrate their dynamical behaviorin the complex plane.

Keywords: Kung–Traub conjecture; multipoint iterations; nonlinear equation; optimal order;basins of attraction

MSC: 65H05; 65D05; 41A25

1. Introduction

Finding faster and exact roots of scalar nonlinear equations is the most important problemin engineering, scientific computing, and applied mathematics. In general, this is the problem ofsolving a nonlinear equation f (x) = 0. Analytical methods for finding solutions of such problemsare almost nonavailable, so the only way to get appropriate solutions by numerical methods isbased on iterative algorithms. Newton’s method is one of the well-known and famous methods forfinding solutions of nonlinear equations or local minima in problems of optimization. Despite its niceproperties, it will often not work efficiently in some real-life practical applications. Ill conditioningof the problems, the computational expense of functional derivative, accurate initial guesses, and alate convergence rate generally lead to difficulties in its use. Nevertheless, many advantages in allof these drawbacks have been found and led to efficient algorithms or codes that can be easily used(see References [1,2] and references therein). Hence, Steffensen developed a derivative-free iterativemethod (SM2) (see References [3]):

w(n) = x(n) + f (x(n)), x(n+1) = x(n) − f (x(n))f [x(n), w(n)]

, (1)

where f [x(n), w(n)] = f (x(n))− f (w(n))

x(n)−w(n) , which preserves the convergence order and efficiency index ofNewton’s method.



The main motivation of this work is to implement efficient derivative-free algorithms for findingthe solution of nonlinear equations. We obtained an optimal iterative method that will support theconjecture [4]. Kung–Traub conjectured that multipoint iteration methods without memory based on dfunctional evaluations could achieve an optimal convergence order 2d−1. Furthermore, we studied thebehavior of iterative schemes in the complex plane.

Let us start a short review of the literature with some of the existing methods with or withoutmemory before proceeding to the proposed idea. Behl et al. [5] presented an optimal scheme thatdoes not need any derivative evaluations. In addition, the given scheme is capable of generating newoptimal eighth-order methods from the earlier optimal fourth-order schemes in which the first sub-stepemploys Steffensen’s or a Steffensen-type method. Salimi et al. [6] proposed a three-point iterativemethod for solving nonlinear equations. The purpose of this work is to upgrade a fourth-order iterativemethod by adding one Newton step and by using a proportional approximation for the last derivative.Salimi et al. [7] constructed two optimal Newton–Secant-like iterative methods for finding solutionsof nonlinear equations. The classes have convergence orders of four and eight and cost only threeand four function evaluations per iteration, respectively. Matthies et al. [8] proposed a three-pointiterative method without memory for solving nonlinear equations with one variable. The methodprovides a convergence order of eight with four function evaluations per iteration. Sharifi et al. [9]presented an iterative method with memory based on the family of King’s methods to solve nonlinearequations. The method has eighth-order convergence and costs only four function evaluations periteration. An acceleration of the convergence speed is achieved by an appropriate variation of a freeparameter in each step. This self-accelerator parameter is estimated using Newton’s interpolationfourth degree polynomial. The order of convergence is increased from eight to 12 without any extrafunction evaluation. Khdhr et al. [10] suggested a variant of Steffensen’s iterative method witha convergence order of 3.90057 for solving nonlinear equations that are derivative-free and havememory. Soleymani et al. [11] presented derivative-free iterative methods without memory withconvergence orders of eight and sixteen for solving nonlinear equations. Soleimani et al. [12] proposeda optimal family of three-step iterative methods with a convergence order of eight by using a weightfunction alongside an approximation for the first derivative. Soleymani et al. [13] gave a class offour-step iterative schemes for finding solutions of one-variable equations. The produced methodshave better order of convergence and efficiency index in comparison with optimal eighth-ordermethods. Soleymani et al. [14] constructed a class of three-step eighth order iterative methods byusing an interpolatory rational function in the third step. Each method of the class reaches the optimalefficiency index according to the Kung–Traub conjecture concerning multipoint iterative methodswithout memory. Kanwar et al. [15] suggested two new eighth-order classes of Steffensen–King-typemethods for finding solutions of nonlinear equations numerically. Cordero et al. [1] proposed a generalprocedure to obtain derivative-free iterative methods for finding solutions of nonlinear equationsby polynomial interpolation. In addition, many authors have worked with these ideas on differentiterative schemes [16–24], describing the basin of attraction of some well-known iterative scheme.In this work, we developed a novel fourth-order iterative scheme, eighth-order iterative scheme,and sixteenth-order iterative scheme, that are without memory, are derivative-free, and are optimal.

The rest of this paper is ordered as follows. In Section 2, we present the proposed fourth-, eighth-,and sixteenth-order methods that are free from derivatives. Section 3 presents the convergence orderof the proposed scheme. In Section 4, we discuss some well-known iterative methods for the numericaland effectiveness comparison of the proposed schemes. In Section 5, we display the performance ofproposed methods and other compared algorithms described by problems. The respective graphicalfractal pictures obtained from each iteration scheme for test problems are given in Section 6 to showthe consistency of the proposed methods. Finally, Section 7 gives concluding remarks.

68


2. Development of Derivative-Free Scheme

2.1. Optimal Fourth-Order Method

Let us start from Steffensen’s method and explain the procedure to get optimal methods ofincreasing order. The idea is to compose a Steffensen’s iteration with Newton’s step as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),

z(n) = y(n) − f (y(n))f ′(y(n)) .

(2)

The resulting iteration has convergence order four, with the composition of two second-ordermethods, but the method is not optimal because it uses four function evaluations. In order to getan optimality, we need to reduce a function and to preserve the same convergence order, and so,we estimate f ′(y(n)) by the following polynomial:

N2(t) = f (y(n)) + (t− y(n)) f [y(n), w(n)] + (t− y(n))(t− w(n)) f [y(n), w(n), x(n)], (3)

where

f [x(0), x(1), x(2), ..., x(k−1), x(k)] =f [x(1), x(2), ..., x(k−1), x(k)]− f [x(0), x(1), x(2), ..., x(k−1)]

x(k) − x(0), x(k) �= x(0),

is the generalized divided differences of kth order at x(0) ≤ x(1) ≤ x(2) ≤ ... ≤ x(k−1) ≤ x(k). It is notedthat N2(y(n)) = f (y(n)). Differentiating Equation (3) and putting t = y(n), we get

N ′2(y

(n)) = f [y(n), w(n)] + (y(n) − w(n)) f [y(n), w(n), x(n)]. (4)

Now, approximating f ′(y(n)) ≈ N ′2(y

(n)) in Equation (2), we get a new derivative-free optimalfourth-order method (PM4) given by⎧⎪⎪⎪⎨⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),

z(n) = y(n) − f (y(n))f [y(n) ,w(n)]+(y(n)−w(n)) f [y(n) ,w(n) ,x(n)]

.

(5)

2.2. Optimal Eighth-Order Method

Next, we attempt to get a new optimal eighth-order method in the following way:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),


,

p(n) = z(n) − f (z(n))f ′(z(n)) .

(6)

The above has eighth-order convergence with five function evaluations, but this is not an optimalmethod. To get an optimal, we need to reduce a function and to preserve the same convergence order,and so, we estimate f ′(z(n)) by the following polynomial:

N3(t) = f (z(n)) + (t− z(n)) f [z(n), y(n)] + (t− z(n))(t− y(n)) f [z(n), y(n), w(n)]

+ (t− z(n))(t− y(n))(t− w(n)) f [z(n), y(n), w(n), x(n)].(7)

69


It is clear that N3(z(n)) = f (z(n)). Differentiating Equation (7) and setting t = z(n), we get

N ′3(z

(n)) = f [z(n), y(n)] + (z(n) − y(n)) f [z(n), y(n), w(n)] + (z(n) − y(n))(z(n) − w(n)) f [z(n), y(n), w(n), x(n)]. (8)

Now, approximating f ′(z(n)) ≈ N ′3(z

(n)) in (6), we get a new derivative-free optimal eighth-ordermethod (PM8) given by⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),


,

p(n) = z(n) − f (z(n))f [z(n) ,y(n)]+(z(n)−y(n)) f [z(n) ,y(n) ,w(n)]+(z(n)−y(n))(z(n)−w(n)) f [z(n) ,y(n) ,w(n) ,x(n)]

.

(9)

2.3. Optimal Sixteenth-Order Method

Next, we attempt to get a new optimal sixteenth-order method in the following way:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),


,


,

x(n+1) = p(n) − f (p(n))f ′(p(n))

.

(10)

The above has sixteenth-order convergence with six function evaluations, but this is not an optimalmethod. To get an optimal, we need to reduce a function and to preserve the same convergence order,and so, we estimate f ′(p(n)) by the following polynomial:

N4(t) = f (p(n)) + (t− p(n)) f [p(n), z(n)] + (t− p(n))(t− z(n)) f [p(n), z(n), y(n)]

+ (t− p(n))(t− z(n))(t− y(n)) f [p(n), z(n), y(n), w(n)]

+ (t− p(n))(t− z(n))(t− y(n))(t− w(n)) f [p(n), z(n), y(n), w(n), x(n)].

(11)

It is clear that N4(p(n)) = f (p(n)). Differentiating Equation (11) and setting t = p(n), we get

N ′4(p(n)) = f [p(n), z(n)] + (p(n) − z(n)) f [p(n), z(n), y(n)] + (p(n) − z(n))(p(n) − y(n)) f [p(n), z(n), y(n), w(n)]

+ (p(n) − z(n))(p(n) − y(n))(p(n) − w(n)) f [p(n), z(n), y(n), w(n), x(n)].(12)

Now, approximating f ′(p(n)) ≈ N ′4(p(n)) in Equation (10), we get a new derivative-free optimal

sixteenth-order iterative method (PM16) given by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

w(n) = x(n) + f (x(n))3,

y(n) = x(n) − f (x(n))4

f (w(n))− f (x(n)),


,


,

x(n+1) = p(n) − f (p(n))N ′

4(p(n)),

(13)

where N ′4(p(n)) given in Equation (12).

70



In this part, we will derive the convergence analysis of the proposed schemes in Equations (5), (9),and (13) with the help of MATHEMATICA software.

Theorem 1. Let f : D ⊂ R→ R be a sufficiently smooth function having continuous derivatives. If f (x) hasa simple root x∗ in the open interval D and x(0) is chosen in a sufficiently small neighborhood of x∗, then themethod of Equation (5) is of local fourth-order convergence and and it satisfies the error equation

en+1 = (c[2]3 − c[2]c[3])e4n + O(e5

n).

Proof. Let en = x(n) − x∗ and c[j] = f (j)(x∗)j! f ′(x∗) , j = 2, 3, 4, .... Expanding f (x(n)) and f (w(n)) about x∗ by

Taylor’s method, we have

f (x(n)) = f ′(x∗)[en + c[2]e2n + c[3]e3

n + c[4]e4n + . . .], (14)

w(n) = en + f ′(x∗)3[en + c[2]e2n + c[3]e3

n + c[4]e4n + . . .]3, (15)

f (w(n)) = f ′(x∗)[en + c[2]e2n + ( f ′(x∗)3 + c[3])e3

n + (5 f ′(x∗)3c[2] + c[4])e4n + . . .]. (16)

Then, we have

y(n) = x∗ + c[2]e2n + (−2c[2]2 + 2c[3])e3

n + (4c[2]3 − 7c[2]c[3] + 3c[4] + f ′(x∗)3c[2])e4n + . . . . (17)

Expanding f (y(n)) about x∗, we have

f (y(n)) = f ′(x∗)[c[2]e2n − 2(c[2]2 − c[3])e3

n + (5c[2]3 − 7c[2]c[3] + 3c[4] + f ′(x∗)3c[2])e4n + . . .]. (18)

Now, we get the Taylor’s expansion of f [y(n), w(n)] = f (y(n))− f (w(n))

y(n)−w(n) by replacing

Equation (15)–(18).

f [y(n), w(n)] = f ′(x∗)[1 + c[2]en + (c[2]2 + c[3])e2n + ( f ′(x∗)3c[2]− 2c[2]3 + c[2]c[3] + c[4])e3

n + . . .]. (19)

Also, we have

f [y(n), w(n), x(n)] = f ′(x∗)[c[2] + 2c[3]en + (c[2]c[3] + c[4])e2n + . . .] (20)

Using Equations (14)–(20) in the scheme of Equation (5), we obtain the following error equation:

en+1 = (c[2]3 − c[2]c[3])e4n + . . . . (21)

This reveals that the proposed method PM4 attains fourth-order convergence.

Theorem 2. Let f : D ⊂ R→ R be a sufficiently smooth function having continuous derivatives. If f (x) hasa simple root x∗ in the open interval D and x(0) is chosen in a sufficiently small neighborhood of x∗, then themethod of Equation (9) is of local eighth-order convergence and and it satisfies the error equation

en+1 = c[2]2(c[2]2 − c[3])(c[2]3 − c[2]c[3] + c[4])e8n + O(e9

n).

71


Theorem 3. Let f : D ⊂ R→ R be a sufficiently smooth function having continuous derivatives. If f (x) hasa simple root x∗ in the open interval D and x(0) is chosen in a sufficiently small neighborhood of x∗, then themethod of Equation (13) is of local sixteenth-order convergence and and it satisfies the error equation

en+1 = c[2]4(

c[2]2 − c[3])2(

c[2]3 − c[2]c[3] + c[4])(

c[2]4 − c[2]2c[3] + c[2]c[4]− c[5])

e16n + O(e17

n ).

4. Some Known Derivative-Free Methods

Let us consider the following schemes for the purpose of comparison. Derivative-freeKung–Traub’s two-step method (KTM4) [4] is as follows:

y(n) = x(n) − f (x(n))f [x(n), w(n)]

, w(n) = x(n) + f (x(n)), x(n+1) = y(n) − f (y(n)) f (w(n))

[ f (w(n))− f (y(n))] f [x(n), y(n)]. (22)

Derivative-free Argyros et al. two-step method (AKKB4) [25] is as follows:

y(n) = x(n) − f (x(n))f [x(n), w(n)]

, w(n) = x(n) + f (x(n)), x(n+1) = y(n) − f (y(n))[ f (x(n))− 2 f (y(n))]

f (x(n))f [y(n), w(n)]

(1− f (y(n))

f (x(n))

). (23)

Derivative-free Zheng et al. two-step method (ZLM4) [26] is as follows:

y(n) = x(n) − f (x(n))f [x(n), w(n)]

, w(n) = x(n) + f (x(n)), x(n+1) = y(n) − f (y(n))f [x(n), y(n)] + (y(n) − x(n)) f [x(n), w(n), y(n)]

. (24)

Derivative-free Argyros et al. three-step method (AKKB8) [25] is as follows:

⎧⎨⎩ y(n) = x(n) − f (x(n))f [x(n) ,w(n)]

, w(n) = x(n) + f (x(n)), z(n) = y(n) − f (y(n))[ f (x(n))−2 f (y(n))]

f (x(n))f [y(n) ,w(n)]

(1− f (y(n))

f (x(n))

),

x(n+1) = z(n) − f (z(n))f [z(n) ,y(n)]+(z(n)−y(n)) f [z(n) ,y(n) ,x(n)]+(z(n)−y(n))(z(n)−x(n)) f [z(n) ,y(n) ,x(n) ,w(n)]

.(25)

Derivative-free Kanwar et al. three-step method (KBK8) [15] is as follows:⎧⎪⎨⎪⎩y(n) = x(n) − f (x(n))

f [x(n) ,w(n)], w(n) = x(n) + f (x(n))3, z(n) = y(n) − f (y(n))

2 f [y(n) ,x(n)]− f [x(n) ,w(n)],

x(n+1) = z(n) − f (z(n))f [y(n) ,z(n)]+ f [w(n) ,y(n) ,z(n)](z(n)−y(n))

(1−

(f (y(n))f (x(n))

)3 − 8 f (y(n)) f (z(n))f (x(n))2 + f (z(n))

f (x(n))+ 5

(f (z(n))f (y(n))

)2).

(26)

Derivative-free Soleymani three-step method (SM8) [2] is as follows:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩w(n) = x(n) + f (x(n)), y(n) = x(n) − f (x(n))

f [x(n) ,w(n)], z(n) = y(n) − f (y(n))

f [x(n) ,w(n)]φn,

x(n+1) = z(n) − f (z(n))f [x(n) ,w(n)]

φnψn, where φn = 11− f (y(n))/ f (x(n))− f (y(n))/ f (w(n))

,

ψn = 1 + 11+ f [x(n) ,w(n)]

(f (y(n))f (x(n))

)2+((1 + f [x(n), w(n)])(2 + f [x(n), w(n)])

)(f (y(n))f (w(n))

)3+ f (z(n))

f (y(n))+ f (z(n))

f (x(n))+ f (z(n))

f (w(n)).

(27)

Derivative-free Zheng et al. four-step method (ZLM16) [26] is as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

y(n) = x(n) − f (x(n))2

f (w(n))− f (x(n)), w(n) = x(n) + f (x(n)), z(n) = y(n) − f (y(n))

f [y(n) ,w(n)]+(y(n)−w(n)) f [y(n) ,w(n) ,x(n)],


,

x(n+1) = p(n) − f (p(n))f ′(p(n))

,

where f ′(p(n)) ≈ f [p(n), z(n)] + (p(n) − z(n)) f [p(n), z(n), y(n)] + (p(n) − z(n))(p(n) − y(n)) f [p(n), z(n), y(n), w(n)]

+(p(n) − z(n))(p(n) − y(n))(p(n) − w(n)) f [p(n), z(n), y(n), w(n), x(n)].

(28)

5. Test Problems

We compare the performance of the proposed methods along with some existing methods fortest problems by using MATLAB. We use the conditions for stopping criteria for | f (x(N))| < ε where

72


ε = 10−50 and N is the number of iterations needed for convergence. The computational order ofconvergence (coc) is given by

ρ =ln |(x(N) − x(N−1))/(x(N−1) − x(N−2))|

ln |(x(N−1) − x(N−2))/(x(N−2) − x(N−3))| .

The test problems and their roots are given below:

f1(x) = sin(2 cos x)− 1− x2 + esin(x3), x∗ = −0.7848959876612125352...

f2(x) = x3 + 4x2 − 10, x∗ = 1.3652300134140968457...

f3(x) =√

x2 + 2x + 5− 2 sin x− x2 + 3, x∗ = 2.3319676558839640103...

f4(x) = e−x sin x + log (1 + x2)− 2, x∗ = 2.4477482864524245021...

f5(x) = sin(x) + cos(x) + x, x∗ = −0.4566247045676308244...

Tables 1–5 show the results of all the test functions with a given initial point. The computationalorder of convergence conforms with theoretical order of convergence. If the initial points are close tothe zero, then we obtain less number of iterations with least error. If the initial points are away fromthe zero, then we will not obtained the least error. We observe that the new methods in all the testfunction have better efficiency as compared to other existing methods of the equivalent methods.

Table 1. Comparisons between different methods for f1(x) at x(0) = −0.9.

Methods N |x(1) − x(0)| |x(2) − x(1)| |x(3) − x(2)| |x(N) − x(N−1)| coc

SM2 (1) 8 0.0996 0.0149 6.1109 ×10−4 1.0372×10−89 1.99KTM4 (22) 5 0.1144 6.7948×10−4 3.4668×10−12 5.1591×10−178 4.00AKKB4 (23) 4 0.1147 3.6299×10−4 9.5806×10−14 4.6824×10−52 3.99ZLM4 (24) 5 0.1145 6.1744×10−4 1.5392×10−12 1.3561×10−184 4.00

PM4 (5) 4 0.1150 1.3758×10−4 2.6164×10−16 3.4237×10−63 3.99AKKB8 (25) 3 0.1151 1.2852×10−8 3.7394×10−62 3.7394×10−62 7.70KBK8 (26) 3 0.1151 8.1491×10−8 1.5121×10−56 1.5121×10−56 7.92SM8 (27) 4 0.1151 1.8511×10−6 1.0266×10−43 0 7.99PM8 (9) 3 0.1151 7.1154×10−9 9.3865×10−67 9.3865×10−67 8.02

ZLM16 (28) 3 0.1151 5.6508×10−15 1.4548×10−225 1.4548×10−225 15.82PM16 (13) 3 0.1151 5.3284×10−17 1.2610×10−262 1.2610×10−262 16.01

Table 2. Comparisons between different methods for f2(x) at x(0) = 1.6.


SM2 (1) 12 0.0560 0.0558 0.0520 1.7507×10−83 1.99KTM4 (22) 5 0.2184 0.0163 3.4822×10−6 4.7027×10−79 3.99AKKB4 (23) 33 0.0336 0.0268 0.0171 2.4368×10−52 0.99ZLM4 (24) 5 0.2230 0.0117 4.4907×10−7 3.9499×10−95 3.99

PM4 (5) 5 0.2123 0.0224 2.3433×10−7 4.3969×10−112 4.00AKKB8 (25) 4 0.2175 0.0173 1.2720×10−9 1.0905×10−66 8.00KBK8 (26) D D D D D DSM8 (27) 4 0.2344 4.1548×10−4 9.5789×10−24 7.7650×10−181 7.89PM8 (9) 4 0.2345 2.4307×10−4 4.6428×10−32 8.2233×10−254 8.00

ZLM16 (28) 3 0.2348 2.2048×10−7 1.9633×10−124 1.9633×10−124 15.57PM16 (13) 3 0.2348 2.8960×10−8 1.7409×10−126 1.7409×10−126 17.11

73




SM2 (1) 7 0.3861 0.0180 4.6738×10−05 1.0220×10−82 1.99KTM4 (22) 4 0.3683 2.8791×10−4 1.0873×10−16 2.2112×10−66 3.99AKKB4 (23) 4 0.3683 2.5241×10−4 5.2544×10−17 9.8687×10−68 3.99ZLM4 (24) 4 0.3683 3.1466×10−4 1.7488×10−16 1.6686×10−65 4.00

PM4 (5) 4 0.3683 2.2816×10−4 2.3732×10−17 2.7789×10−69 3.99AKKB8 (25) 3 0.3680 1.7343×10−8 3.8447×10−67 3.8447×10−67 8.00KBK8 (26) 4 0.3680 4.2864×10−5 1.8700×10−38 2.4555×10−305 7.99SM8 (27) 3 0.3680 7.8469×10−8 2.9581×10−61 2.9581×10−61 8.00PM8 (9) 3 0.3680 9.7434×10−9 1.0977×10−69 1.0977×10−69 8.04

ZLM16 (28) 3 0.3680 1.4143×10−16 6.3422×10−240 6.3422×10−240 16.03PM16 (13) 3 0.3680 3.6568×10−17 7.4439×10−274 7.4439×10−274 16.04



SM2 (1) 7 0.4975 0.0500 2.5378×10−4 1.9405×10−73 2.00KTM4 (22) 4 0.2522 1.7586×10−6 1.5651×10−26 9.8198×10−107 3.99AKKB4 (23) 4 0.5489 0.0011 3.8305×10−15 5.5011×10−61 3.99ZLM4 (24) 4 0.5487 9.0366×10−4 1.4751×10−15 1.0504×10−62 3.99

PM4 (5) 4 0.5481 3.0864×10−4 8.0745×10−18 3.7852×10−72 3.99AKKB8 (25) 3 0.5477 5.4938×10−7 4.9628×10−56 4.9628×10−56 8.17KBK8 (26) 3 0.5477 4.1748×10−7 5.8518×10−59 5.8518×10−59 8.47SM8 (27) 3 0.5477 5.4298×10−7 4.1081×10−56 4.1081×10−56 8.18PM8 (9) 3 0.5477 5.8222×10−8 1.1144×10−64 1.1144×10−64 8.13

ZLM16 (28) 3 0.5477 2.7363×10−14 7.2982×10−229 7.2982×10−229 16.13PM16 (13) 3 0.5477 5.6240×10−16 1.9216×10−257 1.9216×10−257 16.11

Table 5. Comparisons between different methods for f5(x) at x(0) = −0.2.


SM2 (1) 7 0.3072 0.0499 6.4255×10−4 4.1197×10−59 2.00KTM4 (22) 5 0.2585 0.0019 1.5538×10−12 3.4601×10−194 4.00AKKB4 (23) 4 0.2571 4.4142×10−4 3.4097×10−15 1.2154×10−59 3.99ZLM4 (24) 4 0.2580 0.0013 3.5840×10−13 1.8839×10−51 3.99

PM4 (5) 4 0.2569 2.8004×10−4 6.2960×10−17 1.6097×10−67 3.99AKKB8 (25) 3 0.2566 4.1915×10−8 6.3444×10−65 6.3444×10−65 8.37KBK8 (26) 4 0.2566 4.0069×10−6 5.1459×10−47 0 7.99SM8 (27) 4 0.2566 2.9339×10−6 1.0924×10−46 0 7.99PM8 (9) 3 0.2566 3.7923×10−11 9.0207×10−90 9.0207×10−90 7.99

ZLM16 (28) 3 0.2566 5.3695×10−16 7.0920×10−252 7.0920×10−252 16.06PM16 (13) 3 0.2566 1.1732×10−19 1.2394×10−314 1.2394×10−314 16.08

74


6. Basins of Attraction

The iterative scheme gives information about convergence and stability by studying basins ofattraction of the rational function. The basic definitions and dynamical concepts of rational function canfound in References [17,27,28]. Let us consider a region R×R = [−2, 2]× [−2, 2] with 256× 256 grids.We test iterative methods in all the grid point z(0) in the square. The iterative algorithms attempt rootsz∗j of the equation with condition | f (z(k))| < ×10−4 and a maximum of 100 iterations; we conclude

that z(0) is in the basin of attraction of this zero. If the iterative method starting in z(0) reaches a zero inN iterations, then we mark this point z(0) with colors if |z(N) − z∗j | < ×10−4. If N > 50, then we assigna dark blue color for diverging grid points. We describe the basins of attraction for finding complexroots of p1(z) = z2 − 1, p2(z) = z3 − 1, p3(z) = (z2 + 1)(z2 − 1), and p4(z) = z5 − 1 for proposedmethods and some higher-order iterative methods.

In Figures 1–5, we have given the basins of attraction for new methods with some existingmethods. We confirm that a point z0 containing the Julia set whenever the dynamics of point showssensitivity to the conditions. The neighbourhood of initial points leads to the slight variation inbehavior after some iterations. Therefore, some of the compared algorithms obtain more divergentinitial conditions.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

p1(z) p2(z)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

p3(z) p4(z)

Figure 1. Basins of attraction for SM2 for the polynomial.

75


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(a) (b)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(c) (d)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(e) (f)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(g) (h)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(i) (j)

Figure 2. Polynomiographs of p1(z): (a) KTM4; (b) AKKB4; (c) ZLM4; (d) PM4; (e) AKKB8; (f) KBK8;(g) SM8; (h) PM8; (i) ZLM16; and (j) PM16.

76


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(a) (b)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(c) (d)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(e) (f)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(g) (h)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(i) (j)

Figure 3. Polynomiographs of p2(z): (a) KTM4; (b) AKKB4; (c) ZLM4; (d) PM4; (e) AKKB8; (f) KBK8;(g) SM8; (h) PM8; (i) ZLM16; and (j) PM16.

77


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(a) (b)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(c) (d)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(e) (f)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(g) (h)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(i) (j)

Figure 4. Polynomiographs of p3(z): (a) KTM4; (b) AKKB4; (c)ZLM4; (d) PM4; (e) AKKB8; (f) KBK8;(g) SM8; (h) PM8; (i) ZLM16; and (j) PM16.

78


(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

Figure 5. Polynomiographs of p4(z): (a) KTM4; (b) AKKB4; (c)ZLM4; (d) PM4; (e) AKKB8; (f) KBK8;(g) SM8; (h) PM8; (i) ZLM16; and (j) PM16.

79


7. Concluding Remarks

We have proposed fourth-, eighth-, and sixteenth-order methods using finite differenceapproximations. Our proposed new methods requires 3 functions to get the 4th-order method,4 functions to obtain the 8th-order method, and 5 functions to get the 16th-order one. We haveincreased the convergence order of the proposed method, respectively, to four, eight, and sixteenwith efficiency indices 1.587, 1.565, and 1.644 respectively. Our new proposed schemes are betterthan the Steffensen method in terms of efficiency index (1.414). Numerical solutions are tested toshow the performance of the proposed algorithms. Also, we have analyzed on the complex region foriterative methods to study their basins of attraction. Hence, we conclude that the proposed methodsare comparable to other well-known existing equivalent methods.

Author Contributions: Conceptualization, K.M.; Funding acquisition, J.L. and X.W.; Methodology, K.M.; Projectadministration, J.L. and X.W.; Resources, J.L. and X.W.; Writing—original draft, K.M.


Acknowledgments: The authors would like to thank the editors and referees for the valuable comments and forthe suggestions to improve the readability of the paper.


References

1. Cordero, A.; Hueso, J.L.; Martinez, E.; Torregrosa, J.R. Generating optimal derivative free iterative methodsfor nonlinear equations by using polynomial interpolation. Appl. Math. Comp. 2013, 57, 1950–1956.[CrossRef]

2. Soleymani, F. Efficient optimal eighth-order derivative-free methods for nonlinear equations. Jpn. J. Ind.Appl. Math. 2013, 30, 287–306. [CrossRef]

3. Steffensen, J.F. Remarks on iteration. Scand. Aktuarietidskr. 1933, 16, 64–72. [CrossRef]4. Kung, H.; Traub, J. Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 1974,

21, 643–651. [CrossRef]5. Behl, R.; Salimi, M.; Ferrara, M.; Sharifi, S.; Alharbi, S.K. Some Real-Life Applications of a Newly Constructed

Derivative Free Iterative Scheme. Symmetry 2019, 11, 239. [CrossRef]6. Salimi, M.; Long, N.M.A.N.; Sharifi, S.; Pansera, B.A. A multi-point iterative method for solving nonlinear

equations with optimal order of convergence. Jpn. J. Ind. Appl. Math. 2018, 35, 497–509. [CrossRef]7. Salimi, M.; Lotfi, T.; Sharifi, S.; Siegmund, S. Optimal Newton-Secant like methods without memory for

solving nonlinear equations with its dynamics. Int. J. Comput. Math. 2017, 94, 1759–1777. [CrossRef]8. Matthies, G.; Salimi, M.; Sharifi, S.; Varona, J.L. An optimal three-point eighth-order iterative method

without memory for solving nonlinear equations with its dynamics. Jpn. J. Ind. Appl. Math. 2016, 33, 751–766.[CrossRef]

9. Sharifi, S.; Siegmund, S.; Salimi, M. Solving nonlinear equations by a derivative-free form of the King’sfamily with memory. Calcolo 2016, 53, 201–215. [CrossRef]

10. Khdhr, F.W.; Saeed, R.K.; Soleymani, F. Improving the Computational Efficiency of a Variant of Steffensen’sMethod for Nonlinear Equations. Mathematics 2019, 7, 306. [CrossRef]

11. Soleymani, F.; Babajee, D.K.R.; Shateyi, S.; Motsa, S.S. Construction of Optimal Derivative-Free Techniqueswithout Memory. J. Appl. Math. 2012, 2012, 24. [CrossRef]

12. Soleimani, F.; Soleymani, F.; Shateyi, S. Some Iterative Methods Free from Derivatives and Their Basins ofAttraction for Nonlinear Equations. Discret. Dyn. Nat. Soc. 2013, 2013, 10. [CrossRef]

13. Soleymani, F.; Sharifi, M. On a General Efficient Class of Four-Step Root-Finding Methods. Int. J. Math.Comp. Simul. 2011, 5, 181–189.

14. Soleymani, F.; Vanani, S.K.; Paghaleh, M.J. A Class of Three-Step Derivative-Free Root Solvers with OptimalConvergence Order. J. Appl. Math. 2012, 2012, 15. [CrossRef]

15. Kanwar, V.; Bala, R.; Kansal, M. Some new weighted eighth-order variants of Steffensen-King’s type familyfor solving nonlinear equations and its dynamics. SeMA J. 2016. [CrossRef]

80


16. Amat, S.; Busquier, S.; Plaza, S. Dynamics of a family of third-order iterative methods that do not requireusing second derivatives. Appl. Math. Comput. 2004, 154, 735–746. [CrossRef]

17. Amat, S.; Busquier, S.; Plaza, S. Review of some iterative root-finding methods from a dynamical point ofview. SCIENTIA Ser. A Math. Sci. 2004, 10, 3–35.

18. Babajee, D.K.R.; Madhu, K. Comparing two techniques for developing higher order two-point iterativemethods for solving quadratic equations. SeMA J. 2019, 76, 227–248. [CrossRef]

19. Cordero, A.; Hueso, J.L.; Martinez, E.; Torregrosa, J.R. A family of iterative methods with sixth and seventhorder convergence for nonlinear equations. Math. Comput. Model. 2010, 52, 1490–1496. [CrossRef]

20. Curry, J.H.; Garnett, L.; Sullivan, D. On the iteration of a rational function: computer experiments withNewton’s method. Commun. Math. Phys. 1983, 91, 267–277. [CrossRef]

21. Soleymani, F.; Babajee, D.K.R.; Sharifi, M. Modified Jarratt Method Without Memory With Twelfth-OrderConvergence. Ann. Univ. Craiova Math. Comput. Sci. Ser. 2012, 39, 21–34.

22. Tao, Y.; Madhu, K. Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided DifferenceTechniques and Their Basins of Attraction and Its Application. Mathematics 2019, 7, 322. [CrossRef]

23. Vrscay, E.R. Julia sets and mandelbrot-like sets associated with higher order Schroder rational iterationfunctions: a computer assisted study. Math. Comput. 1986, 46, 151–169.

24. Vrscay, E.R.; Gilbert, W.J. Extraneous fxed points, basin boundaries and chaotic dynamics for Schroder andKonig rational iteration functions. Numer. Math. 1987, 52, 1–16. [CrossRef]

25. Argyros, I.K.; Kansal, M.; Kanwar, V.; Bajaj, S. Higher-order derivative-free families of Chebyshev-Halleytype methods with or without memory for solving nonlinear equations. Appl. Math. Comput. 2017,315, 224–245. [CrossRef]

26. Zheng, Q.; Li, J.; Huang, F. An optimal Steffensen-type family for solving nonlinear equations. Appl. Math.Comput. 2011, 217, 9592–9597. [CrossRef]

27. Madhu, K. Some New Higher Order Multi-Point Iterative Methods and Their Applications to Differentialand Integral Equation and Global Positioning System. Ph.D. Thesis, Pndicherry University, Pondicherry,India, June 2016.

28. Scott, M.; Neta, B.; Chun, C. Basin attractors for various methods. Appl. Math. Comput. 2011, 218, 2584–2599.[CrossRef]


81

mathematics

Article

A Generic Family of Optimal Sixteenth-OrderMultiple-Root Finders and Their DynamicsUnderlying Purely Imaginary Extraneous Fixed Points

Min-Young Lee 1, Young Ik Kim 1,* and Beny Neta 2

1 Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea; [email protected] Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA; [email protected]* Correspondence: [email protected]; Tel.: +82-41-550-3415

Received: 26 April 2019; Accepted: 18 June 2019; Published: 20 June 2019

Abstract: A generic family of optimal sixteenth-order multiple-root finders are theoreticallydeveloped from general settings of weight functions under the known multiplicity. Special cases ofrational weight functions are considered and relevant coefficient relations are derived in such a waythat all the extraneous fixed points are purely imaginary. A number of schemes are constructed basedon the selection of desired free parameters among the coefficient relations. Numerical and dynamicalaspects on the convergence of such schemes are explored with tabulated computational results andillustrated attractor basins. Overall conclusion is drawn along with future work on a different familyof optimal root-finders.

Keywords: sixteenth-order optimal convergence; multiple-root finder; asymptotic error constant;weight function; purely imaginary extraneous fixed point; attractor basin

MSC: 65H05; 65H99

1. Introduction

Many nonlinear equations governing real-world natural phenomena cannot be solved exactly byvirtue of their intrinsic complexities. It would be certainly an important matter to discuss methodsfor approximating such solutions of the nonlinear equations. The most widely accepted methodunder general circumstances is Newton’s method, which has quadratic convergence for a simple-rootand linear convergence for a multiple-root. Other higher-order root-finders have been developed bymany researchers [1–9] with optimal convergence satisfying Kung–Traub’s conjecture [10]. Severalauthors [10–14] have proposed optimal sixteenth-order simple-root finders, although their applicationsto real-life problems are limited due to the high degree of their algebraic complexities. Optimalsixteenth-order multiple-root finders are hardly found in the literature to the best of our knowledge atthe time of writing this paper. It is not too much to emphasize the theoretical importance of developingoptimal sixteenth-order multiple root-finders as well as to apply them to numerically solve real-worldnonlinear problems.

In order to develop an optimal sixteenth-order multiple-root finders, we pursue a family ofiterative methods equipped with generic weight functions of the form:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

yn = xn −m f (xn)f ′(xn)

,

zn = yn −mQ f (s)f (yn)f ′(xn)

= xn −m[1 + sQ f (s)

] f (xn)f ′(xn)

,

wn = zn −mK f (s, u) f (zn)f ′(xn)

= xn −m[1 + sQ f (s) + suK f (s, u)

] f (xn)f ′(xn)

,

xn+1 = wn −mJf (s, u, v) f (wn)f ′(xn)

= xn −m[1 + sQ f (s) + suK f (s, u) + suvJ f (s, u, v)

] f (xn)f ′(xn)

,

(1)



where s =( f (yn)

f (xn)

)1/m, u =( f (zn)

f (yn)

)1/m, v =( f (wn)

f (zn)

)1/m; Q f : C→ C is analytic [15] in a neighborhood

of 0, K f : C2 → C holomorphic [16,17] in a neighborhood of (0, 0), and J f : C3 → C holomorphicin a neighborhood of (0, 0, 0). Since s, u and v are respectively one-to-m multiple-valued functions,their principal analytic branches [15] are considered. Hence, for instance, it is convenient to treat sas a principal root given by s = exp[ 1

m Log( f (yn)f (xn)

)], with Log( f (yn)f (xn)

) = Log∣∣ f (yn)

f (xn)

∣∣+ i Arg( f (yn)f (xn)

) for

−π < Arg( f (yn)f (xn)

) ≤ π; this convention of Arg(z) for z ∈ C agrees with that of Log[z] command ofMathematica [18] to be employed later in numerical experiments.

The case for m = 1 has been recently developed by Geum–Kim–Neta [19]. Many other existingcases for m = 1 are special cases of (1) with appropriate forms of weight functions Q f , K f , and J f ;for example, the case developed in [10] uses the following weight functions:⎧⎪⎪⎨⎪⎪⎩

Q f (s) = 1(1−s)2 ,

K f (s, u) = 1+(1−u)s2

(1−s)2(1−u)(1−su)2 ,

J f (s, u, v) = −1+2su2(v−1)+s4(u−1)u2(v−1)(uv−1)+s2[uv−1−u3(v2−1)](1−s)2(u−1)(su−1)2(v−1)(uv−1)(suv−1)2 .

(2)

One goal of this paper is to construct a family of optimal sixteenth-order multiple-root finders bycharacterizing the generic forms of weight functions Q f (s), K f (s, u), and J f (s, u, v). The other goal isto investigate the convergence behavior by exploring their numerical behavior and dynamics throughbasins of attractions [20] underlying the extraneous fixed points [21] when f (z) = (z− a)m(z− b)m isapplied. In view of the right side of final substep of (1), we can conveniently locate extraneous fixedpoints from the roots of the weight function m[1 + sQ f (s) + suK f (s, u) + suvJ f (s, u, v)].

A motivation undertaking this research is to investigate the local and global characters on theconvergence of proposed family of methods (1). The local convergence of an iterative method forsolving nonlinear equations is usually guaranteed with an initial guess taken in a sufficiently closeneighborhood of the sought zero. On the other hand, effective information on its global convergenceis hardly achieved under general circumstances. We can obtain useful information on the globalconvergence from attractor basins through which relevant dynamics is worth exploring. Especiallythe dynamics underlying the extraneous fixed points (to be described in Section 3) would influencethe dynamical behavior of the iterative methods by the presence of possible attractive, indifferent,repulsive, and other chaotic orbits. One way of reducing such influence is to control the location of theextraneous fixed points. We prefer the location to be the imaginary axis that divides the entire complexplane into two symmetrical half-planes. The dynamics underlying the extraneous fixed points on theimaginary axis would be less influenced by the presence of the possible periodic or chaotic attractors.

The main theorem is presented in Section 2 with required constraints on weight functions, Q f ,K f , and J f to achieve the convergence order of 16. Section 2 discusses special cases of rationalweight functions. Section 3 extensively investigates the purely imaginary extraneous fixed points andinvestigates their stabilities. Section 4 presents numerical experiments as well as the relevant dynamics,while Section 5 states the overall conclusions along with the short description of future work.

2. Methods and Special Cases

A main theorem on the convergence of (1) is established here with the error equation andrelationships among generic weight functions Q f (s), K f (s, u), and J f (s, u, v):

Theorem 1. Suppose that f : C → C has a multiple root α of multiplicity m ≥ 1 and is analytic in a

neighborhood of α. Let cj = m!(m−1+j)!

f (m−1+j)(α)f (m)(α)

for j = 2, 3, · · · . Let x0 be an initial guess selected ina sufficiently small region containing α. Assume L f : C → C is analytic in a neighborhood of 0. Let

Qi = 1i!

di

dsi Q f (s)∣∣(s=0) for 0 ≤ i ≤ 6. Let K f : C2 → C be holomorphic in a neighborhood of (0, 0).

Let J f : C3 → C be holomorphic in a neighborhood of (0, 0, 0). Let Kij = 1i!j!

∂i+j

∂si∂uj K f (s, u)∣∣(s=0,u=0) for

83


0 ≤ i ≤ 12 and 0 ≤ j ≤ 6. Let Jijk = 1i!j!k!

∂i+j+k

∂si∂uj∂vj J f (s, u, v)∣∣(s=0,u=0,v=0) for 0 ≤ i ≤ 8, 0 ≤ j ≤ 4 and

0 ≤ k ≤ 2. If Q0 = 1, Q1 = 2, K00 = 1, K10 = 2, K01 = 1, K20 = 1 + Q2, K11 = 4, K30 = −4 + 2Q2 + Q3,J000 = 1, J100 = 2, J200 = 1 + Q2, J010 = 1, J110 = 4, J300 = −4 + 2Q2 + Q3, J001 = 1, J020 = K02, J210 =

1 + K21, J400 = K40, J101 = 2, J120 = 2 + K12, J310 = −4 + K31 + 2Q2, J500 = K50, J011 = 2, J201 =

1 + Q2, J030 = −1 + K02 + K03, J220 = 1 + K21 + K22 − Q2, J410 = −3 + K40 + K41 + Q2 − Q4, J600 =

K60, J111 = 8, J301 = −4 + 2Q2 + Q3, J130 = −4 + 2K02 + K12 + K13, J320 = −6 + 2K21 + K31 + K32 −2Q2 − Q3, J510 = 6 + 2K40 + K50 + K51 − 3Q3 − 2Q4 − Q5, J700 = K70 are fulfilled, then Scheme (1)leads to an optimal class of sixteenth-order multiple-root finders possessing the following error equation: withen = xn − α for n = 0, 1, 2, · · · ,

en+1 =1

3456m15 c2(ρc22 − 2mc3)

[β0c4

2 + β1c22c3 + 12m2(K02 − 1)c2

3 − 12m2c2c4]Ψ e16

n + O(e17n ), (3)

where ρ = 9+m− 2Q2, β0 = (−431+ 12K40− 7m2 + 6m(−17+Q2)+ 102Q2− 24Q3− 12Q4 + 6K21ρ+

3K02ρ2), β1 = −12m(−17+ K21− 2m + Q2 + K02ρ), Ψ = Δ1c82 + Δ2c6

2c3 + Δ3c52c4 + Δ4c3

2c3c4 + Δ5c42 +

Δ6c22 + Δ7c4

3 + Δ8c2c23c4,

Δ1 = (−255124 + 144J800 − 144K80 − 122577m− 23941m2 − 2199m3 − 79m4 + 24K40(472 + 93m +

5m2)− 72(17 + m)Q32 − 576Q2

3 + Q3(48(−566 + 6K40 − 117m− 7m2)− 576Q4) + 24(−485 + 6K40 −108m − 7m2)Q4 − 144Q2

4 + Q22(36(−87 + 14m + m2) + 288Q3 + 144Q4) + Q2(18(5300 + 1529m +

172m2 + 7m3 − 8K40(18 + m)) + 144(35 + m)Q3 + 72(29 + m)Q4) + 18ρ3σ + 6ρ(12J610 − 12(2K50 +

K60 + K61 − 2Q5 − Q6) + J211(−12K40 + σ2) + 2K21(−J002σ2 + σ3 + 6η0)) + ρ2(36J420 − 36J211K21 +

36J002K221 − 72K31 − 36J021K40 − 36K41 − 36K42 + 3J021σ2 + 6K02(−6J401 + 12J002K40 − J002σ2 + σ3)) +

12J401σ7 + J002σ27 + 9ρ4τ),

Δ2 = m(144(Q32 − 2Q5 − Q6) + 288Q3(−K21 + (39 + 4m) − K02ρ) + 144Q2

2(−2K21 − (7 +

m) − K02ρ) + 144Q4(−K21 + 4(9 + m) − K02ρ) + Q2(144K21(58 + 5m) − 36(1529 − 8K40 + m(302 +

17m))− 288Q3 − 144Q4 + 144K02(38 + 3m)ρ)− 108ρ2σ + 6(40859− 24J610 + 48K50 + 24K60 + 24K61 +

24K40(−31 + J211 − 3m) + m(14864 + m(1933 + 88m)) − 2J211σ8) − 72ρ3τ − 24ρ2(J002σ2 − 6η0) −24K21(1309 + m(267 + 14m) − J002σ8 + 6η0) + ρ(144J211K21 − 144J002K2

21 + 12(−12J420 + 12(2K31 +

J021K40 + K41 + K42)− J021σ4)− 12K02(1781 + m(360 + 19m)− 2J002σ4 + 12η0))),Δ3 = 12m2(6(63 + 5m)Q2 − 48Q3− 24Q4 + 6(J211 + K21− 2J002K21)ρ− (1645− 12J401 + 12(−1 +

2J002)K40 + 372m + 23m2 − 2J002σ2) + 3ρ2σ5),Δ4 = 144m3(−3Q2 + (53− J211 + (−1 + 2J002)K21 + 6m− 2J002ρ2)− ρσ5),Δ5 = 72ρm3c5 + 12m2c2

3((−12K21(J211 − 4(10 + m)) + K02(4778 − 12J401 + m(990 + 52m)) +

3(−1929 + 4J401 + 4J420 − 8K31 + 8K40 − 4K41 − 4K42 − 476m − 31m2 + 2J211(43 + 5m) + J021(115 +

22m+m2)))− 6(−88+ 121J021 + 4J211 + 232K02 + 8K21 +(−10+ 13J021 + 22K02)m+ 2J002(51− 4K21 +

5m))Q2 + 12(1 + J002 + 3J021 + 6K02)Q22 + 18ρσ + 18ρ2τ + Q4(12(−2 + K02) + 12η1) + Q3(24(−2 +

K02) + 24η1)− J021η2 + η1((2165− 12K40 + m(510 + 31m)) + η2)),Δ6 = 72m3(2(−1 + J002)mc2

4 − 2mc3c5 + c33(2Q2(−1 + 6K02)− 2K02(49− J211 + 5m + 2J002ρ3) +

(85− 2J211 − 2J230 + 4K12 − 4K21 + 4J002(K21 − ρ2) + 2K22 + 2K23 + 11m + 2J021ρ3)− 4ρτ)),Δ7 = 144m4(−1 + J002 + J021 + J040− K03− K04 + (2− η1)K02 + J002K2

02), Δ8 = 144m4(−3 + η1 +

(1− 2J002)K02),τ = J040 − J021K02 + J002K2

02 − K03 − K04, ρ2 = 17 + 2m−Q2, ρ3 = −26 + K21 − 3m + 3Q2, σ =

J230− J211K02− 2K12− J021K21 + 2J002K02K21−K22−K23, σ2 = 431+ 7m2− 6m(−17+ Q2)− 102Q2 +

24Q3 + 12Q4, σ3 = 472 + 5m2 + m(93− 6Q2)− 108Q2 + 12Q3 + 6Q4, σ4 = 890 + 13m2 − 231Q2 +

6Q22 − 3m(−69 + 7Q2) + 24Q3 + 12Q4, σ5 = J021 + K02 − 2J002K02, σ6 = −1255 + 6K40 − 288m −

17m2 + 363Q2 + 39mQ2 − 18Q22 − 12Q3 − 6Q4, σ7 = 431− 12K40 + 7m2 − 6m(−17 + Q2)− 102Q2 +

24Q3 + 12Q4, σ8 = 1349 + 19m2 + m(312− 36Q2) − 360Q2 + 12Q22 + 24Q3 + 12Q4, η0 = −J401 +

2J002K40, η1 = 2J002 + J021, η2 = 6K221 − 6K21(43 + 5m) + 2σ6K02.

84


Proof. Since Scheme (1) employs five functional evaluations, namely, f ′(xn), f (xn), f (yn), f (zn),and f (wn), optimality can be achieved if the corresponding convergence order is 16. In order to inducethe desired order of convergence, we begin by the 16th-order Taylor series expansion of f (xn) about α:

f (xn) =f ′(α)m!

emn {1 +

17

∑i=2

ci ei−1n + O(e17

n )}. (4)

It follows that

f ′(xn) =f ′(α)

(m− 1)!em−1

n {1 +16

∑i=2

im + i− 1

mci ei−1

n + O(e16n )}. (5)

For brevity of notation, we abbreviate en as e. Using Mathematica [18], we find:

yn = xn −mf (xn)

f ′(xn)= α +

c2

me2 +

(−(m + 1)c22 + 2mc3)

m2 e3 +Y4

m3 e4 +16

∑i=5

Yi

mi−1 ein + O(e17), (6)

where Y4 = (1 + m)2c32 −m(4 + 3m)c2c3 + 3m2c4 and Yi = Yi(c2, c3, · · · , c16) for 5 ≤ i ≤ 16.

After a lengthy computation using the fact that f (yn) = f (xn)|en→(yn−α), we get:

s =(

f (yn)

f (xn)

)1/m

=c2

me +

(−(m + 2)c22 + 2mc3)

m2 e2 +γ3

2m3 e3 +15

∑i=4

Ei ei + O(e16), (7)

where γ3 = (7 + 7m + 2m2)c32 − 2m(7 + 3m)c2c3 + 6m2c4, Ei = Ei(c2, c3, · · · , c16) for 4 ≤ i ≤ 15.

In the third substep of Scheme (1), wn = O(e8) can be achieved based on Kung–Traub’s conjecture.To reflect the effect on wn from zn in the second substep, we need to expand zn up to eighth-orderterms; hence, we carry out a sixth-order Taylor expansion of Q f (s) about 0 by noting that s = O(e)

and f (yn)f ′(xn)

= O(e2):

Q f (s) = Q0 + Q1s + Q2s2 + Q3s3 + Q4s4 + Q5s5 + Q6s6 + O(e7), (8)

where Qj =1j!

dj

dsj Q f (s) for 0 ≤ j ≤ 6. As a result, we come up with:

zn = xn −mQ f (s)f (yn)

f ′(xn)= α +

(1−Q0)

me2 +

μ3

m2 e3 +16

∑i=4

Wi ei + O(e17),

where μ3 = (−1+ m(Q0− 1) + 3Q0−Q1)c22− 2m(Q0− 1)c3 and Wi = Wi(c2, c3, · · · , c16, Q0, · · · , Q6)

for 4 ≤ i ≤ 16. Selecting Q0 = 1 and Q1 = 2 leads us to an expression:

zn = α +c2(ρc2

2 − 2mc3)

m2 e4 +16

∑i=5

Wi ei + O(e17). (9)

By a lengthy computation using the fact that f (zn) = f (xn)|en→(zn−α), we deduce:

u =

(f (zn)

f (yn)

)1/m

=(ρc2

2 − 2mc3)

2m2 e2 +δ3

3m3 e3 +16

∑i=4

Gi ei + O(e17), (10)

where δ3 = (49 + 2m2 + m(27 − 6Q2) − 18Q2 + 3Q3)c32 − 6mρc2c3 + 6m2c4 and Gi =

Gi(c2, c3, · · · , c16, Q2, · · · , Q6) for 4 ≤ i ≤ 16.In the last substep of Scheme (1), xn+1 = O(e16) can be achieved based on Kung-Traub’s conjecture.

To reflect the effect on xn+1 from wn in the third substep, we need to expand wn up to sixteenth-order

85


terms; hence, we carry out a 12th-order Taylor expansion of K f (s, u) about (0, 0) by noting that:

s = O(e), u = O(e2) and f (zn)f ′(xn)

= O(e4) with Kij = 0 satisfying i + 2j > 12 for all 0 ≤ i ≤ 12, 0 ≤ j ≤ 6:

K f (s, u) = K00 + K10s + K20s2 + K30s3 + K40s4 + K50s5 + K60s6 + K70s7 + K80s8 + K90s9 + K100s10 + K110s11+

K120s12 + (K01 + K11s + K21s2 + K31s3 + K41s4 + K51s5 + K61s6 + K71s7 + K81s8 + K91s9 + K101s10)u+(K02 + K12s + K22s2 + K32s3 + K42s4 + K52s5 + K62s6 + K72s7 + K82s8)u2+

(K03 + K13s + K23s2 + K33s3 + K43s4 + K53s5 + K63s6)u3+

(K04 + K14s + K24s2 + K34s3 + K44s4)u4 + (K05 + K15s + K25s2)u5 + K06u6 + O(e13).

(11)

Substituting zn, f (xn), f (yn), f (zn), f ′(xn), and K f (s, u) into the third substep of (1) leads us to:

wn = zn −mK f (s, u) · f (zn)

f ′(xn)= α +

(1− K00)c2(ρc22 − 2mc3)

2m3 e4 +16

∑i=5

Γi ei + O(e17), (12)

where Γi = Γi(c2, c3, · · · , c16, Q2, · · · , Q6, Kj�), for 5 ≤ i ≤ 16, 0 ≤ j ≤ 12 and 0 ≤ � ≤ 6. Thus K00 = 1immediately annihilates the fourth-order term. Substituting K00 = 1 into Γ5 = 0 and solving for K10,we find:

K10 = 2. (13)

Continuing the algebraic operations in this manner at the i-th (6 ≤ i ≤ 7) stage with knownvalues of Kj�, we solve Γi = 0 for remaining Kj� to find:

K20 = 1 + Q2, K01 = 1. (14)

Substituting K00 = 1, K10 = 2, K20 = 1 + Q2, K01 = 1 into (12) and simplifying we find:

v =

(f (wn)

f (zn)

)1/m

= −[β0c4

2 + β1c22c3 + 12m2(K02 − 1)c2

3 − 12m2c2c4]

12m4 e4 +16

∑i=5

Ti ei + O(e17), (15)

where β0 and β1 are described in (3) and Ti = Ti(c2, c3, · · · , c16, Q2, · · · , Q6) for 5 ≤ i ≤ 16.To compute the last substep of Scheme (1), it is necessary to have an eighth-order Taylor expansion

of J f (s, u, v) about (0, 0, 0) due to the fact that f (wn)f ′(xn)

= O(e8). It suffices to expand J f up to eighth-,

fourth-, and second-order terms in s, u, v in order, by noting that s = O(e), u = O(e2), v = O(e4) withJijk = 0 satisfying i + 2j + 4k > 8 for all 0 ≤ i ≤ 8, 0 ≤ j ≤ 4, 0 ≤ k ≤ 2:

J f (s, u, v) = J000 + J100s + J200s2 + J300s3 + J400s4 + J500s5 + J600s6 + J700s7 + J800s8 + (J010 + J110s + J210s2+

J310s3 + J410s4 + J510s5 + J610s6)u + (J020 + J120s + J220s2 + J320s3 + J420s4)u2 + (J030 + J130s + J230s2)u3+

J040u4 + (J001 + J101s + J201s2 + J301s3 + J401s4 + (J011 + J111s + J211s2)u + J021u2)v + J002v2.(16)

Substituting wn, f (xn), f (yn), f (zn), f (wn), f ′(xn) and J f (s, u, v) in (1), we arrive at:

xn+1 = wn −mJf (s, u, v) · f (wn)

f ′(xn)= α + φe8 +

16

∑i=9

Ωi ei + O(e17), (17)

where φ = 124m7 (1 − J000)c2(ρc2

2 − 2mc3)[β0c4

2 + β1c22c3 + 12m2(K02 − 1)c2

3 − 12m2c2c4]

andΩi =Ωi(c2, c3, · · · , c16, Q2, · · · , Q6, Kδθ , Jjk�), for 9 ≤ i ≤ 16, 0 ≤ δ ≤ 12, 0 ≤ θ ≤ 6, 0 ≤ j ≤ 8,0 ≤ k ≤ 4, 0 ≤ � ≤ 2.

Since J000 = 1 makes φ = 0, we substitute J000 = 1 into Ω9 = 0 and solve for J100 to find:

J100 = 2. (18)

86


Continuing the algebraic operations in the same manner at the i-th (10 ≤ i ≤ 15) stage withknown values of Jjk�, we solve Ωi = 0 for remaining Jjk� to find:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

J200 = 1 + Q2, J010 = 1, J110 = 4, J300 = −4 + 2Q2 + Q3, J001 = 1, J020 = K02, J210 = 1 + K21,J400 = K40, J101 = 2, J120 = 2 + K12, J310 = −4 + K31 + 2Q2, J500 = K50, J011 = 2, J201 = 1 + Q2,J111 = 8, J030 = −1 + K02 + K03, J220 = 1 + K21 + K22 −Q2, J410 = −3 + K40 + K41 + Q2 −Q4,J301 = −4 + 2Q2 + Q3, J130 = −4 + 2K02 + K12 + K13, J320 = −6 + 2K21 + K31 + K32 − 2Q2 −Q3,J600 = K60, J510 = 6 + 2K40 + K50 + K51 − 3Q3 − 2Q4 −Q5, J700 = K70.

(19)

Upon substituting Relation (19) into Ω16, we finally obtain:

Ω16 =1

3456m15 c2(ρc22 − 2mc3)

[β0c4

2 + β1c22c3 + 12m2(K02 − 1)c2

3 − 12m2c2c4]Ψ, (20)

where ρ, β0, β1, and Ψ as described in (3). This completes the proof.

Remark 1. Theorem 1 clearly reflects the case for m = 1 with the same constraints on weight functionsQ f , K f , J f studied in [19].

Special Cases of Weight Functions

Theorem 1 enables us to obtain Q f (s), K f (s, u), and J f (s, u, v) by means of Taylor polynomials:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Q f (s) = 1 + 2s + Q2s2 + Q3s3 + Q4s4 + Q5s5 + Q6s6 + O(e7),

K f (s, u) = 1 + 2s + (1 + Q2)s2 + (2Q2 + Q3 − 4)s3 + K40s4 + K50s5 + K60s6 + K70s7 + K80s8

+K90s9 + K100s10 + K110s11 + K120s12 + (1 + 4s + K21s2 + K31s3 + K41s4 + K51s5 + K61s6

+K71s7 + K81s8 + K91s9 + K101s10)u + (K02 + K12s + K22s2 + K32s3 + K42s4 + K52s5

+K6s6 + K72s7 + K82s8)u2 + (K03 + K13s + K23s2 + K33s3 + K43s4 + K53s5 + K63s6)u3

+(K04 + K14s + K24s2 + K34s3 + K44s4)u4 + (K05 + K15s + K25s2)u5 + K06u6 + O(e13),

J f (s, u, v) = 1 + 2s + (1 + Q2)s2 + (2Q2 + Q3 − 4)s3 + K40s4 + K50s5 + K60s6 + K70s7 + J800s8

+(1 + 4s + (1 + K21)s2 + (K31 + 2Q2 − 4)s3 + (K40 + K41 − 3 + Q2 −Q4)s4 + (2K40 + K50 + K51 + 6−3Q3 − 2Q4 −Q5)s5 + J610s6)u + (K02 + (2 + K12)s + (K21 + K22 −Q2 + 1)s2 + (2K21 + K31 + K32 − 6−2Q2 −Q3)s3 + J420s4)u2 + (K02 + K03 − 1 + (2K02 + K12 + K13 − 4)s + J230s2)u3 + J040u4

+(1 + 2s + (1 + Q2)s2 + (2Q2 + Q3 − 4)s3 + J401s4 + (2 + 8s + J211s2)u + J021u2)v + J002v2 + O(e9),

(21)

where parameters Q2–Q6, K40, K50, K60, K70, K80, K90, K100, K110, K120, K21, K31, K41, K51, K61, K71, K81,K91, K101, K02, K12, K22, K32, K42, K52, K62, K72, K82, K03, K13, K23, K33, K43, K53, K63, K04, K14, K24, K34, K44,K05, K15, K25, K06 and J040, J002, J021, J211, J230, J401, J420, J610, J800 may be free.

Although various forms of weight functions Q f (s), K f (s, u) and J f (s, u, v) are available, in thecurrent study we limit ourselves to all three weight functions in the form of rational functions, leadingus to possible purely imaginary extraneous fixed points when f (z) = (z2 − 1)m is employed. In thecurrent study, we will consider two special cases described below:

The first case below will represent the best scheme, W3G7, studied in [19] only for m = 1.Case 1: ⎧⎪⎪⎨⎪⎪⎩

Q f (s) = 11−2s ,

K f (s, u) = Q f (s) · (s−1)2

1−2s−u+2s2u ,

J f (s, u, v) = K f (s, u) · 1+∑3i=1 qisi+u ∑8

i=4 qisi−4+u2 ∑14i=9 qisi−9+u3 ∑21

i=15 qisi−15

A1(s,u)+v·(∑25i=22 risi−22+u(r26+r27s+λs2)

,

(22)

87


where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

q9 = q12 = q13 = q17 = q18 = q19 = q20 = r9 = r10 = r19 = r20 = 0,q1 = −3055820263252−76497245λ

142682111242 , q2 = 56884034112404+44614515451λ285364222484 ,

q3 = −45802209949332−44308526471λ142682111242 , q4 = − 3(17778426888128+67929066997λ)

1426821112420 ,q5 = 2(21034820227211+132665343294λ)

356705278105 , q6 = −1589080655012451+134087681464λ142682111242 ,

q7 = 2(−780300304419180+71852971399λ)71341055621 , q8 = 12288(−727219117761+128167952λ)

71341055621 ,q10 = 2, q11 = 2(−741727036224277+126275739062λ)

71341055621 ,q14 = − 8192(−3964538065856+615849113λ)

71341055621 , q15 = 8(−226231159891830+34083208621λ)71341055621 ,

q16 = − 24(−908116719056544+136634733499λ)356705278105 , q21 = 131072(−918470889768+136352293λ)

356705278105 ,r1 = q1, r2 = q2, r3 = q3, r4 = q4, r5 = q5, r6 = q6 − 1, r7 = q7 − q1 − 2, r8 = q8 +

q32 ,

r11 = −29558910226378916+5256346708371λ1426821112420 , r12 = −55018830261476−109759858153λ

142682111242 ,r13 = 25(−75694849962572+11301475999λ)

71341055621 , r14 = − 4096(−1500792372416+228734011λ)15508925135 ,

r15 = q15, r16 = 43641510974266076−6354680006961λ713410556210 , r17 = − 2(−1060205894022116+202907726307λ)

71341055621 ,r18 = 2(−2870055173156756+475573395275λ)

71341055621 , r21 = q212 , r22 = −1, r23 = −q1, r24 = −q2,

r25 = −q3, r26 = −1− q4, r27 = −2− q1 − q5, λ = 1353974063793787212746858830 ,

(23)

and A1(s, u) = 1 + ∑3i=1 risi + u ∑8

i=4 risi−4 + u2 ∑14i=9 risi−9 + u3 ∑21

i=15 risi−15.As a second case, we will consider the following set of weight functions:

Case 2: ⎧⎪⎪⎨⎪⎪⎩Q f (s) = 1

1−2s ,

K f (s, u) = Q f (s) · (s−1)2

1−2s−u+2s2u ,

J f (s, u, v) =1+∑3

i=1 qisi+u ∑8i=4 qisi−4+u2 ∑14

i=9 qisi−9+u3 ∑19i=15 qisi−15

A0(s,u)+v·(∑23i=20 risi−20+u ∑28

i=24 risi−24+r29u2),

(24)

where A0(s, u) = 1 + ∑3i=1 risi + u ∑8

i=4 risi−4 + u2 ∑14i=9 risi−9 + u3 ∑19

i=15 risi−15 and determination of the 48coefficients qi, ri of J f is described below. Relationships were sought among all free parametersof J f (s, u, v), giving us a simple governing equation for extraneous fixed points of the proposed familyof methods (1).

To this end, we first express s, u and v for f (z) = (z2 − 1)m as follows with t = z2:

s =14(1− 1

t), u =

14· (t− 1)2

(t + 1)2 , v =(t− 1)4

4(1 + 6t + t2)2 . (25)

In order to obtain a simple form of J f (s, u, v), we needed to closely inspect how it is connectedwith K f (s, u). When applying to f (z) = (z2 − 1)m, we find K f (s, u) with t = z2 as shown below:

K f (s, u) =4t(1 + t)

t2 + 6t + 1. (26)

Using the two selected weight functions Q f , K f , we continue to determine coefficients qi, ri ofJ f yielding a simple governing equation for extraneous fixed points of the proposed methods whenf (z) = (z2 − 1)m is applied. As a result of tedious algebraic operations reflecting the 25 constraints(with possible rank deficiency) given by (18) and (19), we find only 23 effective relations, as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

q1 = 14 (−8− r23), q2 = −3− 2q1, q3 = 2 + q1, q4 = −r24, q5 = 2q4 − r25,

q6 = 5 + 7q34 + 13q4

4 + 9q58 − 5q9

4 − 25q108 + 5q12

8 + r84 − 5r11

4 − 5r128 ,

q7 = − 4q35 + 4q4

5 + q55 − 2q6

5 − 2r85 , q8 = q4 +

q52 − q7

2 , q9 = q15 − r15,q10 = −2q4 − 2q15 + q16 − r16, q11 = −6 + q5

2 − 5q72 + q9 + 2q10 − r8 + r11,

r1 = −2 + q1, r2 = 2(1 + q2), r3 = 4q3, r4 = −1 + q4, r5 = 2− q3 − 2q4 + q5,r6 = 1 + 2q3 − q4 − 2q5 + q6, r7 = −2 + 5q3 − 4q4 − 2q5 + 6q7 + 2r8, r9 = −q4 + q9,r10 = −2− q5 − 2q9 + q10, r20 = −1, r21 = 4− q3, r22 = −4(1− q3).

(27)

88


The three relations, J500 = K50, J600 = K60, and J700 = K70 give one relation r22 = −4(1− q3).

Due to 23 constraints in Relation (27), we find that 18 free parameters among 48 coefficients ofJ f in (24) are available. We seek relationships among the free parameters yielding purely imaginaryextraneous fixed points of the proposed family of methods when f (z) = (z2 − 1)m is applied.

To this end, after substituting the 23 effective relations given by (27) into J f in (24) and by applyingto f (z) = (z2 − 1)m, we can construct H(z) = 1 + sQ f (s) + suK f (s, u) + suvJ f (s, u, v) in (1) and seekits roots for extraneous fixed points with t = z2:

H(z) =A · G(t)

t(1 + t)2(1 + 6t + t2) ·W(t), (28)

whereA is a constant factor, G(t) = ∑20i=0 giti, with g0 = −q14, g1 = −16− 2q12− 4q13− 8q14− 16q15 +

10q16 − 4r8 + 4r11 + 2r12 − 1614 − 4r15 − 10r16 + 3r20 + 60r21 + 10r22, gi = gi(q12, r12, · · · , r25), for 2 ≤i ≤ 20 and W(t) = ∑15

i=0 witi, with w0 = −r14, w1 = 16r8 + 4r13 − 5r14 + 4r25, wi =

wi(q12, r12, · · · , r25), for 2 ≤ i ≤ 15. The coefficients of both polynomials, G(t) and W(t), containat most 18 free parameters.

We first observe that partial expressions of H(z) with t = z2, namely, 1 + sQ f (s) = 1+3t2(1+t) , 1 +

sQ f (s)+ suK f (s, u) = 1+21t+35t2+7t3

4(1+t)(1+6t+t2)and the denominator of (28) contain factors t, (1+ 3t), (1+ t), (1+

6t + t2), (1 + 21t + 35t2 + 7t3) when f (z) = (z2 − 1)m is applied. With an observation of presence ofsuch factors, we seek a special subcase in which G(t) may contain all the interested factors as follows:

G(t) = t(1 + 3t)(1 + t)λ(1 + 6t + t2)β(1 + 21t + 35t2 + 7t3) · (1 + 10t + 5t2)(1 + 92t + 134t2 + 28t3 + t4) ·Φ(t), (29)

where Φ(t) is a polynomial of degree (9− (λ+ 2β)), with λ ∈ {0, 1, 2}, β ∈ {1, 2, 3} and 1 ≤ λ+ β ≤ 3;two polynomial factors (1 + 10t + 5t2) and (1 + 92t + 134t2 + 28t3 + t4) were found in Case 3G of theprevious study done by Geum–Kim–Neta [19]. Notice that factors (1 + 6t + t2), (1 + 21t + 35t2 + 7t3),(1 + 10t + 5t2) and (1 + 92t + 134t2 + 28t3 + t4) of G(t) are all negative, i.e., the correspondingextraneous fixed points are all purely imaginary.

In fact, the degree of Φ(t) will be decreased by annihilating the relevant coefficientscontaining free parameters to make all its roots negative. We take the 6 pairs of (λ, β) ∈{(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (2, 1)} to form 6 subcases named as Case 2A–2F in order. The lengthyalgebraic process eventually leads us to additional constraints to each subcase described below:Case 2A: (λ, β) = (0, 1)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q12

q15

r8

r11

r12

r13

r14

r15

r16

r18

r23

r24

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−3 9 9 132

72 −13 0 0 −12 − 23

2 −13 −1340 − 1

2 − 14 − 1

8 − 116 0 0 0 0 0 0 0

− 74

174 5 59

16 2 − 14

116 − 181

28 − 699112 − 99

16 − 1807224 − 1163

16− 11

4 34 1714

1254

1598 −3 3

4 − 2917 − 274

7 − 752 − 629

14 −454− 11

2 14 232

354

92 1 − 1

4 − 25514 − 425

28 −14 − 82156 − 693

47 −30−33− 49

2 − 292 1 − 1

2297

753714 37 1199

289372

0 1 1 34

12 0 0 − 9

7 − 1514 −1 − 27

28 − 272

12 − 21

4 − 234 − 67

16 − 218

14 − 1

1616528

599112

7916

1291224

96716

− 32 16 35

2514 8 −1 1

4 − 24114 − 439

28 − 292 − 947

56 − 7074

2 −21−23− 674 − 21

2 0 −1 1527

27914

372

60328

4472

0 0 0 0 0 0 0 − 27 − 4

7 −1 − 127 0

0 0 0 0 0 0 0 − 27 − 1

14 0 128 − 1

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q13

q16

q17

q18

q19

r17

r19

r25

r26

r27

r28

r29

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−740

− 220756

− 19007

− 135514

19297− 61

71987

56− 1453

14919716727

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

and q14 = 0. These 12 additional constraints q12, q15, r8, r11, r12, r13, r14, r15, r16, r18, r23, r24 are expressedin terms of 12 parameters q13, q16, q17, q18, q19, r17, r19, r25, r26, r27, r28, r29 that are arbitrarily free for

89


the purely imaginary extraneous fixed points. Those 12 free parameters are chosen at our disposal.Then, using Relations (27) and (30), the desired form of J f (s, u, v) in (24) can be constructed.Case 2B: (λ, β) = (0, 2)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q12

q15

r8

r11

r12

r13

r14

r15

r16

r18

r23

r24

r25

r26

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−3 9 9 132

72 0 0 1

6 − 13 − 178

30 − 1

2 − 14 − 1

8 − 116 0 0 0 0 0

− 74

174 5 59

16 2 − 14

116 − 1

48 − 16196 − 917

24− 11

4 34 1714

1254

1598 −3 3

456 − 25

6 − 6683

− 112 14 23

2354

92 1 − 1

416

6124 − 317

67 −30 −33 − 49

2 − 292 1 − 1

2 0 74 217

0 1 1 34

12 0 0 0 1

4 −512 − 21

4 − 234 − 67

16 − 218

14 − 1

16 − 1148

596

61724

− 32 16 35

2514 8 −1 1

423 − 5

24 − 4556

2 −21 −23 − 674 − 21

2 0 −1 − 56

512

2933

0 0 0 0 0 0 0 − 13 − 4

3 − 43

0 0 0 0 0 0 0 0 14 3

0 0 0 0 0 0 0 13 − 2

3 − 443

0 0 0 0 0 0 0 − 43 − 1

3293

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q13

q16

q17

q18

q19

r17

r19

r27

r28

r29

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−100− 87

8−76

252

57−1438− 33

22304−1612

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(31)

and q14 = 0. These 14 additional constraints are expressed in terms of 10 parametersq13, q16, q17, q18, q19, r17, r19,r27, r28, r29 that are arbitrarily free for the purely imaginary extraneousfixed points. Those 10 free parameters are chosen at our disposal. Then, using Relations (27) and (31),the desired form of J f (s, u, v) in (24) can be constructed.Case 2C: (λ, β) = (0, 3)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q12

q15

q16

r8

r11

r12

r13

r14

r15

r16

r18

r23

r24

r25

r26

r27

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−3 0 − 14 −1 0 0 − 23

16 − 294

0 14

14

316 0 0 1

32 − 218

0 −1 − 34 − 1

2 0 0 − 116

214

− 74

34

12 − 1

8 − 14

116 − 15

8 − 332

− 114

354

234

238 −3 3

4 −9 −20− 11

2 − 52 − 7

4 − 52 1 − 1

498

512

7 −3 −2 12 1 − 1

2298

1192

0 0 0 0 0 0 316

14

12 − 1

2 − 14 0 1

4 − 116

98 − 17

2− 3

232

34 0 −1 1

4 − 278

552

2 −2 −1 0 0 −1 7116 − 147

40 0 0 0 0 0 − 1

4 −110 0 0 0 0 0 1

4 30 0 0 0 0 0 − 7

4 −50 0 0 0 0 0 4 −290 0 0 0 0 0 − 13

4 29

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q13

q17

q18

q19

r17

r19

r28

r29

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

− 574

− 18

14−9−100

192

992− 3

413− 77

22014

134−2964−39

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

and q14 = 0. These 16 additional constraints are expressed in terms of 8 parametersq17, q18, q19, r17, r18, r19, r28, r29 that are arbitrarily free for the purely imaginary extraneous fixed points.Those 8 free parameters are chosen at our disposal. Then, using Relations (27) and (32), the desiredform of J f (s, u, v) in (24) can be constructed.Case 2D: (λ, β) = (1, 1), being identical with Case 2A.Case 2E: (λ, β) = (1, 2), being identical with Case 2B.

90


Case 2F: (λ, β) = (2, 1),⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q12

q15

q16

r8

r11

r12

r13

r14

r15

r16

r18

r23

r24

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−3 0 − 14 −1 0 0 −13 −12 − 23

2 −13 −1340 1

414

316 0 0 0 0 0 0 0

0 −1 − 34 − 1

2 0 0 0 0 0 0 0− 7

434

12 − 1

8 − 14

116 − 181

28 − 699112 − 99

16 − 1807224 − 1163

16− 11

4354

234

238 −3 3

4 − 2917 − 274

7 − 752 − 629

14 −454− 11

2 − 52 − 7

4 − 52 1 − 1

4 − 25514 − 425

28 −14 − 82156 − 693

47 −3 −2 1

2 1 − 12

2977

53714 37 1199

28937

20 0 0 0 0 0 − 9

7 − 1514 −1 − 27

28 − 272

12 − 1

2 − 14 0 1

4 − 116

16528

599112

7916

1291224

96716

− 32

32

34 0 −1 1

4 − 24114 − 439

28 − 292 − 947

56 − 7074

2 −2 −1 0 0 −1 1527

27914

372

60328

4472

0 0 0 0 0 0 − 27 − 4

7 −1 − 127 0

0 0 0 0 0 0 − 27 − 1

14 0 128 − 1

2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

q13

q17

q18

q19

r17

r19

r25

r26

r27

r28

r29

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−7400

− 220756

− 19007

− 135514

19297− 61

7198756

− 145314

919716727

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)

and q14 = 0. These 13 additional constraints are expressed in terms of 11 parameters q13,q17, q18, q19,r17, r19, r25, r26, r27, r28, r29 that are arbitrarily free for the purely imaginary extraneous fixed points.Those 11 free parameters are chosen at our disposal. Then, using Relations (27) and (33), the desiredform of J f (s, u, v) in (24) can be constructed. After a process of careful factorization, we find theexpression for H(z) in (28) stated in the following lemma.

Proposition 1. The expression H(z) in (28) is identical in each subcase of 2A–2F and given by a uniquerelation below:

H(z) = (1+3t)(1+10t+5t2)(1+92t+134t2+28t3+t4)8(1+t)(1+6t+t2)(1+28t+70t2+28t3+t4)

, t = z2, (34)

despite the possibility of different coefficients in each subcase.

Proof. Let us write G(t) in (28) as G(t) = t(1 + 3t) · ψ1(t) · ψ2(t) ·Φ(t) · (1 + t)λ(1 + 6t + t2)β−1 withψ1(t) = (1 + 6t + t2)(1 + 21t + 35t2 + 7t3) and ψ2(t) = (1 + 10t + 5t2)(1 + 92t + 134t2 + 28t3 + t4).Then after a lengthy process of a series of factorizations with the aid of Mathematica symbolic ability,we find Φ(t) and W(t) in each subcase as follows.

(1) Case 2A: with λ = 0 and β = 1, we get{Φ(t) = − 2

7 (1 + t) · Γ1(t),W(t) = − 16

7 ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ1(t),(35)

where Γ1(t) = −244 + 28q16 + 28q17 + 21q18 + 14q19 − 36r25 − 30r26 − 28r27 − 27r28 − 378r29 +

14t(−72+ 4q16 + 4q17 + 3q18 + 2q19− 4r25 + 2r27 + 4r28− 72r29)+ t2(1692− 476q16− 476q17− 357q18−238q19 + 548r25 + 578r26 + 672r27 + 957r28 + 6006r29)+ 4t3(−2288+ 196q16 + 196q17 + 147q18 + 98q19−148r25 − 100r26 − 42r27 − 6r28 − 1540r29)− 7t4(1636 + 68q16 + 68q17 + 51q18 + 34q19 + 4r25 + 22r26 +

36r27 + 55r28 + 386r29) + t5(−4176+ 56q16 + 56q17 + 42q18 + 28q19 + 648r25 + 400r26 + 140r27− 32r28 +

7168r29) + t6(−4332 + 28q16 + 28q17 + 21q18 + 14q19 − 484r25 − 394r26 − 392r27 − 545r28 − 2926r29).

(2) Case 2B: with λ = 0 and β = 2, we get{Φ(t) = − 2

3 (1 + t) · Γ2(t),W(t) = − 16

3 (1 + 6t + t2)ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ2(t),(36)

where Γ2(t) = (1 + 6t + t2)(3(−4 + 4q16 + 4q17 + 3q18 + 2q19 + r28 − 20r29) + t(24− 48q16 − 48q17 −36q18− 24q19 + 4r27 + 22r28 + 280r29) + 6t2(−32+ 12q16 + 12q17 + 9q18 + 6q19 + 2r27 + 6r28− 16r29)−2t3(396 + 24q16 + 24q17 + 18q18 + 12q19 + 2r27 + 11r28 + 140r29) + 3t4(−188 + 4q16 + 4q17 + 3q18 +

2q19 − 4r27 − 13r28 + 52r29)).

91


(3) Case 2C: with λ = 0 and β = 3, we get{Φ(t) = 1

2 (1 + t) · Γ3(t),W(t) = 4(1 + 6t + t2)2ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ3(t),

(37)

where Γ3(t) = (1 + 6t + t2)2(12− 3r28 + 2t(60 + r28 − 84r29)− 4r29 + t2(124 + r28 + 172r29)).

(4) Case 2D: with λ = 1 and β = 1, we get{Φ(t) = − 2

7 · Γ1(t),W(t) = − 16

7 ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ1(t).(38)

(5) Case 2E: with λ = 1 and β = 2, we get{Φ(t) = − 2

3 · Γ2(t),W(t) = − 16

3 (1 + 6t + t2)ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ2(t),(39)

(6) Case 2F: with λ = 2 and β = 1, we get{Φ(t) = 2

7 · Γ4(t),W(t) = 2

7 (1 + t)ψ1(t)(1 + 28t + 70t2 + 28t3 + t4)Γ4(t),(40)

where Γ4(t) = 244 + 36r25 + 30r26 + 28r27 + 27r28 + 378r29 + t(764 + 20r25 − 30r26 − 56r27 − 83r28 +

630r29) − 2t2(1228 + 284r25 + 274r26 + 308r27 + 437r28 + 3318r29) + 2t3(5804 + 580r25 + 474r26 +

392r27 + 449r28 + 6398r29) − t4(156 + 1132r25 + 794r26 + 532r27 + 513r28 + 10094r29) + t5(4332 +

484r25 + 394r26 + 392r27 + 545r28 + 2926r29).Substituting each pair of (Φ(t), W(t)) into (28) yields an identical Relation (34) as desired.

Remark 2. The factorization process in the above proposition yields the additional constraints given by (30)–(33)for subcases 2A–2F, after a lengthy computation. Case 2D and Case 2E are found to be identical with Case 2Aand Case 2B, respectively, by direct computation.

In Table 1, we list free parameters selected for typical subcases of 2A–2F. Combining these selectedfree parameters with Relations (27) and (30)–(33), we can construct special iterative schemes namedas W2A1, W2A2, · · · , W2F3, W2F4. Such schemes together with W3G7 for Case 1 shall be used inSection 4 to display results on their numerical and dynamical aspects.

Table 1. Free parameters selected for typical subcases of 2A1–2F4.

SCN q13 q16 q17 q18 q19 r17 r19 r25 r26 r27 r28 r29

2A1 0 0 0 0 0 0 0 0 0 0 0 02A2 20 0 0 0 0 0 1012 4 2 -8 0 02A3 − 89

26 0 0 0 0 0 0 − 14926 0 0 0 0

2A4 0 0 71126 − 622

1322213 0 0 − 149

26 0 0 0 02B1 0 0 0 0 0 0 0 - - 0 0 02B2 0 0 0 0 0 0 96 - - −31 −1 − 1

42B3 −19 0 0 0 0 0 0 - - −18 0 02B4 0 0 45 −52 −12 0 0 - - −18 0 02C1 0 - 0 0 0 0 0 - - - 0 02C2 −34 - 0 0 0 174 0 - - - 4 02C3 0 - 0 0 0 0 280 - - - 4 02C4 − 39

4 - 3754 − 627

4 0 0 0 - - - 0 02F1 0 - 0 0 0 −40 0 − 122

32903 − 184

3 0 02F2 16 - 0 0 0 0 0 1 0 -10 0 02F3 138

7 - − 3567

396314 0 0 0 0 0 0 0 0

2F4 0 - 0 0 0 −32 0 −33 78 −46 −4 0

92


3. The Dynamics behind the Extraneous Fixed Points

The dynamics behind the extraneous fixed points [21] of iterative map (1) have been investigatedby Stewart [20], Amat et al. [22], Argyros–Magreñan [23], Chun et al. [24], Chicharro et al. [25],Chun–Neta [26], Cordero et al. [27], Geum et al. [14,19,28–30], Rhee at al. [9], Magreñan [31],Neta et al. [32,33], and Scott et al. [34].

We locate a root α of a given function f (x) as a fixed point ξ of the iterative map R f :

xn+1 = R f (xn), n = 0, 1, · · · , (41)

where R f is the iteration function associated with f . Typically, R f is written in the form: R f (xn) =

xn − f (xn)f ′(xn)

Hf (xn), where Hf is a weight function whose zeros are other fixed points ξ �= α calledextraneous fixed points of R f . The dynamics of R f might be influenced by presence of possibleattractive, indifferent, or repulsive, and other periodic or chaotic orbits underlying the extraneousfixed points. For ease of analysis, we rewrite the iterative map (41) in a more specific form:

xn+1 = R f (xn) = xn −mf (xn)

f ′(xn)Hf (xn), (42)

where Hf (xn) = 1 + sQ f (s) + suK f (s, u) + suvJ f (s, u, v) can be regarded as a weight function in theclassical modified Newton’s method for a multiple root of integer multiplicity m. Notice that α is afixed point of R f , while ξ �= α for which Hf (ξ) = 0 are extraneous fixed points of R f .

The influence of extraneous fixed points on the convergence behavior was well demonstratedfor simple zeros via König functions and Schröder functions [21] applied to a class of functions{ fk(x) = xk − 1, k ≥ 2}. The basins of attraction may be altered due to the trapped sequence {xn}by the attractive extraneous fixed points of R f . An initial guess x0 chosen near a desired root mayconverge to another unwanted remote root when repulsive or indifferent extraneous fixed points arepresent. These aspects of the Schröder functions were observed when applied to the same class offunctions { fk(x) = xk − 1, k ≥ 2}.

To simply treat dynamics underlying the extraneous fixed points of iterative map (42), we select amember f (z) = (z2 − 1)m. By a similar approach made by Chun et al. [35] and Neta et al. [33,36], weconstruct Hf (xn) = s ·Q f (s) + s · u · K f (s, u) + s · u · v · J f (s, u, v) in (42). Applying f (z) = (z2 − 1)m

to Hf , we find a rational function H(z) with t = z2:

H(z) =N (t)D(t)

, (43)

where both D(t) and N (t) are co-prime polynomial functions of t. The underlying dynamics of theiterative map (42) can be favorably investigated on the Riemann sphere [37] with possible fixed points“0(zero)” and “∞”. As can be seen in Section 5, the relevant dynamics will be illustrated in a 6× 6square region centered at the origin.

Indeed, the roots t ofN (t) provide the extraneous fixed points ξ of R f in Map (42) by the relation:

ξ =

{t

12 , if t �= 0,

0(double root), if t = 0.(44)

Extraneous Fixed Points and their Stability

The following proposition describes the stability of the extraneous fixed points of (42).

93


Proposition 2. Let f (z) = (z2 − 1)m. Then the extraneous fixed points ξ for Case 2 discussed earlier are allfound to be repulsive.

Proof. By direct computation of R′f (z) with f (z) = (z2 − 1)m, we write it as with t = z2:

R′f (z) =Ψn(t)Ψd(t)

,

where Ψn(t) = (−1 + t)15 and Ψd(t) = 16t(1 + t)2(1 + 6t + t2)2(1 + 28t + 70t2 + 28t3 + t4)2.With the help of Mathematica, we are able to express Ψn(t) = 1

61509375 (1 + 3t)(1 + 10t + 5t2)(1 +

92t + 134t2 + 28t3 + t4) ·Qn(t)− 2097152 · Rn(t) and Ψd(t) = − 161509375 16(1 + 3t)(1 + 10t + 5t2)(1 +

92t + 134t2 + 28t3 + t4) · Qd(t) − 131072 · Rd(t), with Qn(t) and Qd(t) as six- and eight-degreepolynomials, while Rn(t) = (327, 923, 929, 643 + 34, 417, 198, 067, 010t + 446, 061, 306, 116, 505t2 +

1621107643125740t3 + 2, 036, 953, 856, 667, 405t4 + 892, 731, 761, 917, 554t5 + 108, 873, 731, 877, 775t6)

and Rd(t) = (327, 923, 929, 643 + 34417198067010t + 446, 061, 306, 116, 505t2 + 1621107643125740t3 +

2, 036, 953, 856, 667, 405t4 + 892, 731, 761, 917, 554t5 + 108, 873, 731, 877, 775t6). Further, we expressRn(t) = (1 + 10t + 5t2)Qν(t) + Rν(t) and Rd(t) = (1 + 10t + 5t2)Qδ(t) + Rδ(t), with Rν(t) =

− 10,077,69625 (36 + 341t) = Rδ(t). Now let t = ξ2, then

R′f (ξ) = 16

using the fact that (1 + 3t)(1 + 10t + 5t2)(1 + 92t + 134t2 + 28t3 + t4) = 0. Hence ξ for Case 2 are allfound to be repulsive.

Remark 3. Although not described here in detail due to limited space, by means of a similar proof as shown inProposition 2, extraneous fixed points ξ for Case 1 was found to be indifferent in [19].

If f (z) = p(z) is a generic polynomial other than (z2− 1)m, then theoretical analysis of the relevantdynamics may not be feasible as a result of the highly increased algebraic complexity. Nevertheless,we explore the dynamics of the iterative map (42) applied to f (z) = p(z), which is denoted by Rp asfollows:

zn+1 = Rp(zn) = zn −mp(zn)

p′(zn)Hp(zn). (45)

Basins of attraction for various polynomials are illustrated in Section 5 to observe the complicateddynamics behind the fixed points or the extraneous fixed points. The letter W was convenientlyprefixed to each case number in Table 1 to symbolize a way of designating the numerical and dynamicalaspects of iterative map (42) .

4. Results and Discussion on Numerical and Dynamical Aspects

We first investigate numerical aspects of the local convergence of (1) with schemes W3G7 andW2A1–W2F4 for various test functions; then we explore the dynamical aspects underlying extraneousfixed points based on iterative map (45) applied to f (z) = (z2− 1)m, whose attractor basins give usefulinformation on the global convergence.

Results of numerical experiments are tabulated for all selected methods in Tables 2–4.Computational experiments on dynamical aspects have been illustrated through attractor basins inFigures 1–7. Both numerical and dynamical aspects have strongly confirmed the desired convergence.

Throughout the computational experiments with the aid of Mathematica, $MinPrecision = 400has been assigned to maintain 400 digits of minimum number of precision. If α is not exact, then it isgiven by an approximate value with 416 digits of precision higher than $MinPrecision.

Limited paper space allows us to list xn and α with up to 15 significant digits. We set error boundε = 1

2 × 10−360 to meet |xn − α| < ε. Due to the high-order of convergence and root multiplicity, close

94


initial guesses have been selected to achieve a moderate number of accurate digits of the asymptoticerror constants.

Methods W3G7, W2A1, W2C2 and W2F2 successfully located desired zeros of test functionsF1 − F4: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

W3G7 : F1(x) = [cos (πx2 ) + 2x2 − 3π]4, α ≈ 2.27312045629419, m = 4,

W2A1 : F2(x) = [cos (x2 + 1)− x log (x2 − π + 2) + 1]4 · (x2 + 1− π), α =√

π − 1, m = 5,W2C2 : F3(x) = [sin−1 (x2 − 1) + 3ex − 2x− 3]2, α ≈ 0.477696831914490, m = 2,W2F2 : F4(x) = (x2 + 1)4 + log[1 + (x2 + 1)3], α = i, m = 3,where log z(z ∈ C) is a principal analytic branch with − π < Im(log z) ≤ π.

(46)

We find that Table 2 ensures sixteenth-order convergence. The computational asymptoticerror constant |en|/|en−1|16 is in agreement with the theoretical one η = limn→∞ |en|/|en−1|16

up to 4 significant digits. The computational convergence order pn = log |en/η|/log |en−1| wellapproaches 16.

Additional test functions in Table 3 confirm the convergence of Scheme (1). The errors |xn − α|are listed in Table 4 for comparison among the listed methods W3G7 and W2A1–W2F4. In the currentexperiments, W3G7 has slightly better convergence for f5 and slightly poor convergence for all othertest functions than the rest of the listed methods. No specific method performs better than the otheramong methods W2A1–W2F4 of Case 2.

According to the definition of the asymptotic error constant η(ci, Qf , Kf , J f ) = limn→∞ |Rf (xn)− α|/|xn − α|16, the convergence is dependent on iterative map R f (xn), f (x), x0, α and the weight functionsQ f , K f and J f . It is clear that no particular method always achieves better convergence than the othersfor any test functions.

Table 2. Convergence of methods W3G7, W2A1, W2C2, W2F2 for test functions F1(x)− F4(x).

Method F n xn |F(xn)| |xn − α| |en/e16n−1| η pn

0 2.2735 1.873 × 10−10 0.000379544W3G7 F1 1 2.27312045629419 1.927× 10−233 6.798 × 10−60 0.00003666355445 0.00003666729357

2 2.27312045629419 0.0× 10−400 1.004× 10−237 16.00000

0 1.4634 1.93× 10−21 0.0000181404W2A1 F2 1 1.46341814037882 3.487× 10−366 2.040 × 10−74 148.4148965 148.4575003

2 1.46341814037882 0.0× 10−400 0.0× 10−399 16.00000

0 0.4777 1.890 × 10−10 3.168 × 10−6

W2C2 F3 1 0.477696831914490 6.116 × 10−183 1.802 × 10−92 0.0001750002063 0.00017499998262 0.477696831914490 0.0 × 10−400 8.522 × 10−367 16.00000

0 0.99995i 1.000 × 10−12 0.00005W2F2 F4 1 1.00000000000000 i 4.820× 10−215 1.391× 10−72 0.001037838436 0.001041219259

2 1.00000000000000 i 0.0 × 10−400 0.0× 10−400 16.00030

i =√−1, η = limn→∞

|en ||en−1 |16 , pn =

log |en/η|log |en−1 | .

Table 3. Additional test functions fi(x) with zeros α and initial values x0 and multiplicities.

i fi(x) α x0 m

1 [4 + 3 sin x− 2x2]4 ≈ 1.85471014256339 1.86 42 [2x− Pi + x cos x]5 π

2 1.5707 53 [2x3 + 3e−x + 4 sin (x2)− 5]2 ≈ −0.402282449584416 −0.403 24 [

√3x2 · cos πx

6 + 1x3+1 − 1

28 ] · (x− 3)3 3 3.0005 4

5 (x− 1)2 + 112 − log[ 25

12 − 2x + x2] 1− i√

36 0.99995− 0.28i 2

6 [x log x−√x + x3]3 1 1.0001 3

Here, log z (z ∈ C) represents a principal analytic branch with − π ≤ Im(log z) < π.

The proposed family of methods (1) has efficiency index EI [38], which is 161/5 ≈ 1.741101 andlarger than that of Newton’s method. In general, the local convergence of iterative methods (45) is

95


guaranteed with good initial values x0 that are close to α. Selection of good initial values is a difficulttask, depending on precision digits, error bound, and the given function f (x).

Table 4. Comparison of |xn − α| among selected methods applied to various test functions.

Method |xn − α| f (x)f1 f2 f3 f4 f5 f6

W3G7|x1 − α| 1.77 × 10−40 * 1.62 × 10−57 4.89 × 10−51 1.50 × 10−61 5.76 × 10−7 1.19 × 10−62

|x2 − α| 1.02 × 10−159 1.13 × 10−225 1.24 × 10−201 3.27 × 10−245 1.08 × 10−95 2.40 × 10−247

W2A1|x1 − α| 2.83 × 10−42 1.05 × 10−58 1.92 × 10−52 1.23 × 10−62 1.29 × 10−6 1.11 × 10−63

|x2 − α| 0.0 × 10−399 4.24 × 10−230 0.0 × 10−400 0.0 × 10−399 6.62 × 10−90 3.61 × 10−251

W2A2|x1 − α| 1.63 × 10−41 2.33 × 10−58 1.45 × 10−51 1.05 × 10−61 1.32 × 10−6 2.34 × 10−63

|x2 − α| 0.0 × 10−399 1.11 × 10−228 0.0 × 10−400 0.0 × 10−399 2.53 × 10−89 7.39 × 10−250

W2A3|x1 − α| 2.53 × 10−43 1.82 × 10−60 4.56 × 10−54 1.20 × 10−63 4.43 × 10−6 3.85 × 10−65

|x2 − α| 0.0 × 10−399 1.40 × 10−236 8.40 × 10−213 0.0 × 10−399 3.03 × 10−83 1.53 × 10−256

W2A4|x1 − α| 1.24 × 10−42 1.35 × 10−59 1.53 × 10−52 8.38 × 10−63 1.58 × 10−4 1.12 × 10−64

|x2 − α| 1.70 × 10−125 5.96 × 10−424 5.22 × 10−155 8.16 × 10−187 3.34 × 10−57 2.79 × 10−424

W2B1|x1 − α| 2.39 × 10−42 2.28 × 10−59 1.30 × 10−52 7.80 × 10−63 2.14 × 10−6 2.81 × 10−64

|x2 − α| 0.0 × 10−399 1.57 × 10−232 0.0 × 10−400 0.0 × 10−399 6.04 × 10−87 2.23 × 10−253

W2B2|x1 − α| 4.44 × 10−42 2.73 × 10−59 3.03 × 10−52 1.79 × 10−62 4.30 × 10−6 3.29 × 10−64

|x2 − α| 0.0 × 10−399 6.69 × 10−232 0.0 × 10−400 0.0 × 10−399 6.01 × 10−82 7.78 × 10e−253

W2B3|x1 − α| 9.85 × 10−43 3.11 × 10−61 4.26 × 10−53 3.01 × 10−63 4.46 × 10−6 3.26 × 10−65

|x2 − α| 0.0 × 10−399 1.17 × 10−239 0.0 × 10−400 0.0 × 10−399 3.06 × 10−83 7.91 × 10−257

W2B4|x1 − α| 1.04 × 10−42 1.92 × 10−59 1.77 × 10−52 1.12 × 10−62 1.77 × 10−4 1.53 × 10−64

|x2 − α| 1.12 × 10−125 4.68 × 10−405 9.06 × 10−155 2.23 × 10−186 2.32 × 10−56 0.0 × 10−400

W2C1|x1 − α| 4.87 × 10−42 4.27 × 10−59 2.95 × 10−52 1.50 × 10−62 1.08 × 10−6 5.14 × 10−64

|x2 − α| 0.0 × 10−399 0.0 × 10−399 0.0 × 10−400 0.0 × 10−399 2.41 × 10−91 2.22 × 10−191

W2C2|x1 − α| 9.31 × 10−42 1.01 × 10−58 5.94 × 10−52 2.61 × 10−62 1.47 × 10−6 1.18 × 10−63

|x2 − α| 0.0 × 10−399 4.23 × 10−230 0.0 × 10−400 0.0 × 10−399 7.11 × 10−89 4.95 × 10−251

W2C3|x1 − α| 9.17 × 10−42 8.88 × 10−59 6.85 × 10−52 4.30 × 10−62 4.36 × 10−6 9.66 × 10−64

|x2 − α| 0.0 × 10−399 7.84 × 10−230 0.0 × 10−400 0.0 × 10−399 2.11 × 10−81 6.05 × 10−251

W2C4|x1 − α| 5.36 × 10−42 6.72 × 10−59 6.55 × 10−52 4.38 × 10−62 1.02 × 10−4 6.13 × 10−64

|x2 − α| 9.92 × 10−124 5.96 × 10−424 2.97 × 10−153 8.55 × 10−185 1.30 × 10−59 0.0 × 10−400

W2F1|x1 − α| 4.36 × 10−42 8.67 × 10−60 2.55 × 10−52 1.29 × 10−62 4.34 × 10−6 1.60 × 10−64

|x2 − α| 0.0 × 10−399 7.12 × 10−234 0.0 × 10−400 0.0 × 10−399 4.57 × 10−82 4.61 × 10−254

W2F2|x1 − α| 1.20 × 10−42 1.74 × 10−60 5.52 × 10−53 3.67 × 10−63 4.08 × 10−6 5.06 × 10−65

|x2 − α| 0.0 × 10−399 1.16 × 10−236 0.0 × 10−400 0.0 × 10−399 2.33 × 10−83 4.56 × 10−256

W2F3|x1 − α| 1.08 × 10−41 1.54 × 10−58 1.55 × 10−51 1.14 × 10−61 8.67 × 10−5 1.396 × 10−63

|x2 − α| 1.41 × 10−424 5.27 × 10−172 8.47 × 10−424 0.0 × 10−399 1.66 × 10−60 5.22 × 10−188

W2F4|x1 − α| 3.80 × 10−42 5.01 × 10−60 2.15 × 10−52 1.07 × 10−62 4.35 × 10−6 1.18 × 10−64

|x2 − α| 0.0 × 10−399 1.16 × 10−235 0.0 × 10−400 0.0 × 10−399 3.65 × 10−82 1.38 × 10−254

The global convergence with appropriate initial values x0 is effectively described by means ofa basin of attraction that is the set of initial values leading to long-time behavior approaching theattractors under the iterative action of R f . Basins of attraction contain information about the region ofconvergence. A method occupying a larger region of convergence is likely to be a more robust method.A quantitative analysis will play the important role for measuring the region of convergence.

The basins of attraction, as well as the relevant statistical data, are constructed in a similarmanner shown in the work of Geum–Kim–Neta [19]. Because of the high order, we take a smallersquare [−1.5, 1.5]2 and use 601 × 601 initial points uniformly distributed in the domain. Maplesoftware has been used to perform the desired dynamics with convergence stopping criteria satisfying|xn+1 − xn| < 10−6 within the maximum number of 40 iterations. An initial point is painted with acolor whose intensity measures the number of iterations converging to a root. The brighter color impliesthe faster convergence. The black point means that its orbit did not converge within 40 iterations.

Despite the limited space, we will explore the dynamics of all listed maps W3G7 and W2A1–W2F4,with applications to pk(z), (1 ≤ k ≤ 7) through the following seven examples. In each example,we have shown dynamical planes for the convergence behavior of iterative map xn+1 = R f (xn) (42)

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with f (z) = pk(z) by illustrating the relevant basins of attraction through Figures 1–7 and displayingrelevant statistical data in Tables 5–7 with colored fonts indicating best results.

Example 1. As a first example, we have taken a quadratic polynomial raised to the power of two with all realroots:

p1(z) = (z2 − 1)2. (47)

Clearly the roots are ±1. Basins of attraction for W3G7, W2A1–W2F4 are given in Figure 1. ConsultingTables 5–7, we find that the methods W2B2 and W2F4 use the least number (2.71) of iterations per point onaverage (ANIP) followed by W2F1 with 2.72 ANIP, W2C3 with 2.73 and W2B1 with 2.74. The fastest methodis W2A2 with 969.374 s followed closely by W2A3 with 990.341 s. The slowest is W2A4 with 4446.528 s.Method W2C4 has the lowest number of black points (601) and W2A4 has the highest number (78843). We willnot include W2A4 in the coming examples.

Table 5. Average number of iterations per point for each example (1–7).

MapExample

1: m = 2 2: m = 3 3: m = 3 4: m = 4 5: m = 3 6: m = 3 7: m = 3 Average

W3G7 2.94 3.48 3.83 3.93 3.95 3.97 6.77 4.12W2A1 2.84 3.50 3.70 4.04 6.84 3.74 5.49 4.31W2A2 2.76 3.15 3.52 3.84 3.62 3.66 4.84 3.63W2A3 2.78 3.21 3.61 3.89 3.70 3.74 4.98 3.70W2A4 11.40 - - - - - - -W2B1 2.74 3.25 3.70 3.88 3.67 3.72 5.01 3.71W2B2 2.95 3.42 3.66 4.01 3.75 3.77 5.15 3.82W2B3 2.78 3.28 3.64 3.89 3.69 4.65 5.13 3.86W2B4 3.29 3.91 4.99 - - - - -W2C1 2.88 3.66 3.87 4.08 3.89 5.45 6.25 4.30W2C2 2.93 3.68 3.95 4.15 6.70 4.67 5.75 4.55W2C3 2.73 3.22 3.53 3.98 3.60 3.62 4.94 3.66W2C4 3.14 3.81 4.96 - - - - -W2F1 2.72 3.24 3.55 3.84 3.49 3.57 5.41 3.69W2F2 2.81 3.28 3.80 4.06 5.02 4.50 5.29 4.10W2F3 2.91 3.54 4.36 4.41 - - - -W2F4 2.71 3.19 3.50 3.86 3.42 3.53 5.52 3.68

Example 2. As a second example, we have taken the same quadratic polynomial now raised to the power ofthree:

p2(z) = (z2 − 1)3. (48)

The basins for the best methods are plotted in Figure 2. This is an example to demonstrate the effect ofraising the multiplicity from two to three. In one case, namely W3G7, we also have m = 5 with CPU time of4128.379 s. Based on the figure we see that W2B4, W2C4 and W2F3 were chaotic. The worst are W2B4, W2C4and W2F3. In terms of ANIP, the best was W2A2 (3.15) followed by W2F4 (3.19) and the worst was W2B4(3.91). The fastest was W2B3 using (2397.111 s) followed by W2F1 using 2407.158 s and the slowest was W2C4(4690.295 s) preceded by W3G7 (2983.035 s). Four methods have the highest number of black points (617).Those were W2A1, W2B4, W2C1 and W2F2. The lowest number was 601 for W2A2, W2C2, W2C4 and W2F1.

Comparing the CPU time for the cases m = 2 and m = 3 of W3G7, we find it is about doubled. But whenincreasing from three to five, we only needed about 50% more.

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2A4 (6) W2B1 (7) W2B2 (8) W2B3

(9) W2B4 (10) W2C1 (11) W2C2 (12) W2C3

(13) W2C4 (14) W2F1 (15) W2F2 (16) W2F3

(17) W2F4

Figure 1. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2A4 (left), W2B1 (center left), W2B2 (center right) and W2B3 (right). The thirdrow for W2B4 (left), W2C1 (center left), W2C2 (center right) and W2C3 (right). The third row for W2C4(left), W2F1 (center left), W2F2 (center right), and W2F3 (right). The bottom row for W2F4 (center),for the roots of the polynomial equation (z2 − 1)2.

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2B4

(9) W2C1 (10) W2C2 (11) W2C3 (12) W2C4

(13) W2F1 (14) W2F2 (15) W2F3 (16) W2F4

(17) W3G7m5

Figure 2. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2B4 (right). The thirdrow for W2C1 (left), W2C2 (center left), W2C3 (center right) and W2C4 (right). The fourth row forW2F1 (left), W2F2 (center left), W2F3 (center right), and W2F4 (right). The bottom row for W3G7m5(center), for the roots of the polynomial equation (z2 − 1)k, k ∈ {3, 5}.

Example 3. In our third example, we have taken a cubic polynomial raised to the power of three:

p3(z) = (3z3 + 4z2 − 10)3. (49)

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Basins of attraction are given in Figure 3. It is clear that W2B4, W2C4 and W2F3 were too chaotic andthey should be eliminated from further consideration. In terms of ANIP, the best was W2F4 (3.50) followed byW2A2 (3.52), W2C3 (3.53) and W2F1 (3.55) and the worst were W2B4 and W2C4 with 4.99 and 4.96 ANIP,respectively. The fastest was W2C3 using 2768.362 s and the slowest was W2B3 (7193.034 s). There were 13methods with only one black point and one with two points. The highest number of black points was 101 forW2F2.

Table 6. CPU time (in seconds) required for each example(1–7) using a Dell Multiplex-990.

MapExample

1: m = 2 2: m = 3 3: m = 3 4: m = 4 5: m = 3 6: m = 3 7: m = 3 Average

W3G7 1254.077 2983.035 3677.848 3720.670 3944.937 3901.679 4087.102 3367.050W2A1 1079.667 2694.537 3528.149 3119.911 5896.635 2938.747 3526.840 3254.927W2A2 969.374 2471.727 3287.081 2956.702 3218.223 2891.478 2981.179 2682.252W2A3 990.341 2843.789 2859.093 2999.712 3002.146 3074.811 3155.307 2703.600W2A4 4446.528 - - - - - - -W2B1 1084.752 2634.826 3295.162 3051.941 2835.755 3238.363 3272.667 2773.352W2B2 1075.393 2429.996 3130.223 3051.192 2929.106 3581.456 3155.619 2764.712W2B3 1180.366 2397.111 7193.034 3000.383 2970.711 3739.766 3139.084 3374.351W2B4 1274.653 2932.008 4872.972 - - - - -W2C1 1132.069 2685.355 3242.637 3287.066 3147.663 4080.019 4802.662 3196.782W2C2 1112.162 2881.697 3189.706 3873.037 5211.619 3665.773 3950.896 3412.127W2C3 1052.570 2421.026 2768.362 3014.033 2778.518 2914.941 3953.346 2700.399W2C4 2051.710 4690.295 7193.034 - - - - -W2F1 1071.540 2407.158 2909.965 3472.317 2832.230 3490.896 3246.584 2775.813W2F2 1015.051 2438.483 3031.802 3061.270 3703.152 3737.394 3324.537 2901.670W2F3 1272.188 2596.200 3603.655 4130.158 - - - -W2F4 1216.839 2620.052 3589.177 3233.168 3534.312 3521.660 3934.845 3092.865

Table 7. Number of points requiring 40 iterations for each example (1–7).

MapExample

1: m = 2 2: m = 3 3: m = 3 4: m = 4 5: m = 3 6: m = 3 7: m = 3 Average

W3G7 677 605 1 250 40 1265 33,072 5130.000W2A1 657 617 1 166 34,396 1201 18,939 7996.714W2A2 697 601 1 162 1 1201 15,385 2578.286W2A3 675 605 55 152 9 1221 14,711 2489.714W2A4 78,843 - - - - - - -W2B1 679 613 1 204 9 1201 13,946 2379.000W2B2 635 609 1 116 1 1217 15,995 2653.429W2B3 679 613 1 146 3 10,645 16,342 4061.286W2B4 659 617 2 - - - - -W2C1 689 617 1 400 20 18,157 24,239 6303.286W2C2 669 601 1 174 17,843 1265 18,382 5562.143W2C3 659 609 1 184 1 1201 14,863 2502.571W2C4 601 601 1 - - - - -W2F1 681 601 1 126 10 1225 18,772 3059.429W2F2 679 617 101 614 3515 1593 17,469 3512.571W2F3 663 605 1 78 - - - -W2F4 645 605 1 130 12 1217 20,020 3232.857

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2B4

(9) W2C1 (10) W2C2 (11) W2C3 (12) W2C4

(13) W2F1 (14) W2F2 (15) W2F3 (16) W2F4

Figure 3. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2B4 (right). The thirdrow for W2C1 (left), W2C2 (center left), W2C3 (center right) and W2C4 (right). The bottom row forW2F1 (left), W2F2 (center left), W2F3 (center right), and W2F4 (right), for the roots of the polynomialequation (3z3 + 4z2 − 10)3.

Example 4. As a fourth example, we have taken a different cubic polynomial raised to the power of four:

p4(z) = (z3 − z)4. (50)

The basins are given in Figure 4. We now see that W2F3 is the worst. In terms of ANIP, W2A2 and W2F1were the best (3.84 each) and the worst was W2F3 (4.41). The fastest was W2A2 (2956.702 s) and the slowestwas W2F3 (4130.158 s). The lowest number of black points (78) was for method W2F3 and the highest number(614) for W2F2. We did not include W2F3 in the rest of the experiments.

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2C1

(9) W2C2 (10) W2C3 (11) W2F1 (12) W2F2

(13) W2F3 (14) W2F4

Figure 4. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2C1 (right). The thirdrow for W2C2 (left), W2C3 (center left), W2F1 (center right) and W2F2 (right). The bottom row forW2F3 (left) and W2F4 (right), for the roots of the polynomial equation (z3 − z)4.

Example 5. As a fifth example, we have taken a quintic polynomial raised to the power of three:

p3(z) = (z5 − 1)3. (51)

The basins for the best methods left are plotted in Figure 5. The worst were W2A1 and W2C2. In terms ofANIP, the best was W2F4 (3.42) followed by W2F1 (3.49) and the worst were W2A1 (6.84) and W2C2 (6.70).The fastest was W2C3 using 2778.518 s followed by W2F1 using 2832.23 s and W2B1 using 2835.755 s.The slowest was W2A1 using 5896.635 s. There were three methods with one black point (W2A2, W2B2 andW2C3) and four others with 10 or less such points, namely W2B3 (3), W2A3 and W2B1 (9) and W2F1 (10).The highest number was for W2A1 (34,396) preceded by W2C2 with 17,843 black points.

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2C1

(9) W2C2 (10) W2C3 (11) W2F1 (12) W2F2

(13) W2F4

Figure 5. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2C1 (right). The thirdrow for W2C2 (left), W2C3 (center left), W2F1 (center right) and W2F2 (right). The bottom row forW2F4 (center), for the roots of the polynomial equation (z5 − 1)3.

Example 6. As a sixth example, we have taken a quartic polynomial raised to the power of three:

p6(z) = (z4 − 1)3. (52)

The basins for the best methods left are plotted in Figure 6. It seems that most of the methods left were goodexcept W2B3 and W2C1. Based on Table 5 we find that W2F4 has the lowest ANIP (3.53) followed by W2F1(3.57). The fastest method was W2A2 (2891.478 s) followed by W2C3 (2914.941 s). The slowest was W2C1(4080.019 s) preceded by W3G7 using 3901.679 s. The lowest number of black points was for W2A1, W2A2,W2B1 and W2C3 (1201) and the highest number was for W2C1 with 18,157 black points.

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(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2C1

(9) W2C2 (10) W2C3 (11) W2F1 (12) W2F2

(13) W2F4

Figure 6. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2C1 (right). The thirdrow for W2C2 (left), W2C3 (center left), W2F1 (center right) and W2F2 (right). The bottom row forW2F4 (center), for the roots of the polynomial equation (z4 − 1)3.

Example 7. As a seventh example, we have taken a non-polynomial equation having ±i as its triple roots:

p6(z) = (z + i)3(ez−i − 1)3, with i =√−1. (53)

The basins for the best methods left are plotted in Figure 7. It seems that most of the methods left have alarger basin for the root −i, i.e., the boundary does not match the real line exactly. Based on Table 5 we find thatW2A2 has the lowest ANIP (4.84) followed by W2C3 (4.94) and W2A3 (4.98). The fastest method was W2A2(2981.179 seconds) followed by W2B3 (3139.084 s), W2A3 (3155.307 s) and W2B2 (3155.619 s). The slowestwas W2C1 (4802.662 s). The lowest number of black points was for W2B1 (13,946) and the highest number wasfor W3G7 with 33,072 black points. In general all methods had higher number of black points compared to thepolynomial examples.

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We now average all these results across the seven examples to try and pick the best method.W2A2 had the lowest ANIP (3.63), followed by W2C3 with 3.66, W2F4 with 3.68 and W2F1 with 3.69.The fastest method was W2A2 (2682.252 seconds), followed by W2C3 (2700.399 s) and W2A3 using2703.600 s of CPU. W2B1 has the lowest number of black points on average (2379), followed by W2A3(2490 black points). The highest number of black points was for W2A1.

Based on these seven examples we see that W2F4 has four examples with the lowest ANIP, W2A2had three examples and W2F1 has one example. On average, though, W2A2 had the lowest ANIP.W2A2 was the fastest in four examples and on average. W2C3 was the fastest in two examples andW2B3 in one example. In terms of black points, W2A2, W2B1 and W2B3 had the lowest number inthree examples and W2F1 in two examples. On average W2B1 has the lowest number. Thus, werecommend W2A2, since it is in the top in all categories.

(1) W3G7 (2) W2A1 (3) W2A2 (4) W2A3

(5) W2B1 (6) W2B2 (7) W2B3 (8) W2C1

(9) W2C2 (10) W2C3 (11) W2F1 (12) W2F2

(13) W2F4

Figure 7. The top row for W3G7 (left), W2A1 (center left), W2A2 (center right) and W2A3 (right).The second row for W2B1 (left), W2B2 (center left), W2B3 (center right) and W2C1 (right). The thirdrow for W2C2 (left), W2C3 (center left), W2F1 (center right) and W2F2 (right). The bottom row forW2F4 (center), for the roots of the non-polynomial equation (z + i)3(ez−i − 1)3.

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5. Conclusions

Both numerical and dynamical aspects of iterative map (1) support the main theorem well througha number of test equations and examples. The W2C2 and W2B3 methods were observed to occupyrelatively slower CPU time. Such dynamical aspects would be greatly strengthened if we could includea study of parameter planes with reference to appropriate parameters in Table 1.

The proposed family of methods (1) employing generic weight functions favorably cover most ofoptimal sixteenth-order multiple-root finders with a number of feasible weight functions. The dynamicsbehind the purely imaginary extraneous fixed points will choose best members of the family withimproved convergence behavior. However, due to the high order of convergence, the algebraicdifficulty might arise resolving its increased complexity. The current work is limited to univariatenonlinear equations; its extension to multivariate ones becomes another task.

Author Contributions: investigation, M.–Y.L.; formal analysis, Y.I.K.; supervision, B.N.

Conflicts of Interest: The authors have no conflict of interest to declare.

References

1. Bi, W.; Wu, Q.; Ren, H. A new family of eighth-order iterative methods for solving nonlinear equations.Appl. Math. Comput. 2009, 214, 236–245. [CrossRef]

2. Cordero, A.; Torregrosa, J.R.; Vassileva, M.P. Three-step iterative methods with optimal eighth-orderconvergence. J. Comput. Appl. Math. 2011, 235, 3189–3194. [CrossRef]

3. Geum, Y.H.; Kim, Y.I. A uniparametric family of three-step eighth-order multipoint iterative methods forsimple roots. Appl. Math. Lett. 2011, 24, 929–935. [CrossRef]

4. Lee, S.D.; Kim, Y.I.; Neta, B. An optimal family of eighth-order simple-root finders with weight functionsdependent on function-to-function ratios and their dynamics underlying extraneous fixed points. J. Comput.Appl. Math. 2017, 317, 31–54. [CrossRef]

5. Liu, L.; Wang, X. Eighth-order methods with high efficiency index for solving nonlinear equations.Appl. Math. Comput. 2010, 215, 3449–3454. [CrossRef]

6. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Džunic, J. Multipoint Methods for Solving Nonlinear Equations; Elsevier:New York, NY, USA, 2012.

7. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Džunic, J.; Multipoint methods for solving nonlinear equations:A survey. Appl. Math. Comput. 2014, 226, 635–660. [CrossRef]

8. Sharma, J.R.; Arora, H. A new family of optimal eighth order methods with dynamics for nonlinear equations.Appl. Math. Comput. 2016, 273, 924–933. [CrossRef]

9. Rhee, M.S.; Kim, Y.I.; Neta, B. An optimal eighth-order class of three-step weighted Newton’s methodsand their dynamics behind the purely imaginary extraneous fixed points. Int. J. Comput. Math. 2017, 95,2174–2211. [CrossRef]

10. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 1974, 21,643–651. [CrossRef]

11. Maroju, P.; Behl, R.; Motsa, S.S. Some novel and optimal families of King’s method with eighth andsixteenth-order of convergence. J. Comput. Appl. Math. 2017, 318, 136–148. [CrossRef]

12. Sharma, J.R.; Argyros, I.K.; Kumar, D. On a general class of optimal order multipoint methods for solvingnonlinear equations. J. Math. Anal. Appl. 2017, 449, 994–1014. [CrossRef]

13. Neta, B. On a family of Multipoint Methods for Non-linear Equations. Int. J. Comput. Math. 1981, 9, 353–361.[CrossRef]

14. Geum, Y.H.; Kim, Y.I.; Neta, B. Constructing a family of optimal eighth-order modified Newton-typemultiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points.J. Comput. Appl. Math. 2018, 333, 131–156. [CrossRef]

15. Ahlfors, L.V. Complex Analysis; McGraw-Hill Book, Inc.: New York, NY, USA, 1979.16. Hörmander, L. An Introduction to Complex Analysis in Several Variables; North-Holland Publishing Company:

Amsterdam, The Netherlands, 1973.

106


17. Shabat, B.V. Introduction to Complex Analysis PART II, Functions of several Variables; American MathematicalSociety: Providence, RI, USA, 1992.

18. Wolfram, S. The Mathematica Book, 5th ed.; Wolfram Media: Champaign, IL, USA, 2003.19. Geum, Y.H.; Kim, Y.I.; Neta, B. Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders

and Investigating Their Dynamics. Mathematics 2019, 7, 8. [CrossRef]20. Stewart, B.D. Attractor Basins of Various Root-Finding Methods. Master’s Thesis, Naval Postgraduate

School, Department of Applied Mathematics, Monterey, CA, USA, June 2001.21. Vrscay, E.R.; Gilbert, W.J. Extraneous Fixed Points, Basin Boundaries and Chaotic Dynamics for shröder and

König rational iteration Functions. Numer. Math. 1988, 52, 1–16. [CrossRef]22. Amat, S.; Busquier, S.; Plaza, S. Review of some iterative root-finding methods from a dynamical point of

view. Scientia 2004, 10, 3–35.23. Argyros, I.K.; Magreñán, A.Á. On the convergence of an optimal fourth-order family of methods and its

dynamics. Appl. Math. Comput. 2015, 252, 336–346. [CrossRef]24. Chun, C.; Lee, M.Y.; Neta, B.; Dzunic, J. On optimal fourth-order iterative methods free from second

derivative and their dynamics. Appl. Math. Comput. 2012, 218, 6427–6438. [CrossRef]25. Chicharro, F.; Cordero, A.; Gutiérrez, J.M.; Torregrosa, J.R. Complex dynamics of derivative-free methods for

nonlinear equations. Appl. Math. Comput. 2013, 219, 7023–7035. [CrossRef]26. Chun, C.; Neta, B. Comparison of several families of optimal eighth order methods. Appl. Math. Comput.

2016, 274, 762–773. [CrossRef]27. Cordero, A.; García-Maimó, J.; Torregrosa, J.R.; Vassileva, M.P.; Vindel, P. Chaos in King’s iterative family.

Appl. Math. Lett. 2013, 26, 842–848. [CrossRef]28. Geum, Y.H.; Kim, Y.I.; Magreñán, Á.A. A biparametric extension of King’s fourth-order methods and their

dynamics. Appl. Math. Comput. 2016, 282, 254–275. [CrossRef]29. Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modified

double-Newton type and their dynamics. Appl. Math. Comput. 2015, 270, 387–400. [CrossRef]30. Geum, Y.H.; Kim, Y.I.; Neta, B. A sixth-order family of three-point modified Newton-like multiple-root

finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 2016, 283, 120–140.[CrossRef]

31. Magreñan, Á.A. Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput.2014, 233, 29–38.

32. Neta, B.; Scott, M.; Chun, C. Basin attractors for various methods for multiple roots. Appl. Math. Comput.2012, 218, 5043–5066. [CrossRef]

33. Neta, B.; Chun, C.; Scott, M. Basins of attraction for optimal eighth order methods to find simple roots ofnonlinear equations. Appl. Math. Comput. 2014, 227, 567–592. [CrossRef]

34. Scott, M.; Neta, B.; Chun, C. Basin attractors for various methods. Appl. Math. Comput. 2011, 218, 2584–2599.[CrossRef]

35. Chun, C.; Neta, B.; Basins of attraction for Zhou-Chen-Song fourth order family of methods for multipleroots. Math. Comput. Simul. 2015, 109, 74–91. [CrossRef]

36. Neta, B.; Chun, C. Basins of attraction for several optimal fourth order methods for multiple roots.Math. Comput. Simul. 2014, 103, 39–59. [CrossRef]

37. Beardon, A.F. Iteration of Rational Functions; Springer: New York, NY, USA, 1991.38. Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: Chelsea, VT, USA, 1982.


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Article

An Efficient Family of Optimal Eighth-OrderMultiple Root Finders

Fiza Zafar 1,2,* , Alicia Cordero 2 and Juan R. Torregrosa 2

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University,Multan 60800, Pakistan

2 Instituto de Matemáticas Multidisciplinar, Universitat Politènica de València, 46022 València, Spain;[email protected] (A.C.); [email protected] (J.R.T.)


Received: 13 November 2018 ; Accepted: 5 December 2018; Published: 7 December 2018

Abstract: Finding a repeated zero for a nonlinear equation f (x) = 0, f : I ⊆ R → R has always been ofmuch interest and attention due to its wide applications in many fields of science and engineering.Modified Newton’s method is usually applied to solve this kind of problems. Keeping in viewthat very few optimal higher-order convergent methods exist for multiple roots, we present anew family of optimal eighth-order convergent iterative methods for multiple roots with knownmultiplicity involving a multivariate weight function. The numerical performance of the proposedmethods is analyzed extensively along with the basins of attractions. Real life models from life science,engineering, and physics are considered for the sake of comparison. The numerical experiments anddynamical analysis show that our proposed methods are efficient for determining multiple roots ofnonlinear equations.

Keywords: nonlinear equations; multiple zeros; optimal iterative methods; higher order of convergence

1. Introduction

It is well-known that Newton’s method converges linearly for non-simple roots of a nonlinearequation. For obtaining multiple roots of a univariate nonlinear equation with a quadratic order ofconvergence, Schröder [1] modified Newton’s method with prior knowledge of the multiplicitym ≥ 1 of the root as follows:

xn+1 = xn −mf (xn)

f ′(xn). (1)

Scheme (1) can determine the desired multiple root with quadratic convergence and is optimal inthe sense of Kung-Traub’s conjecture [2] that any multipoint method without memory can reach itsconvergence order of at most 2p−1 for p functional evaluations.

In the last few decades, many researchers have worked to develop iterative methods for findingmultiple roots with greater efficiency and higher order of convergence. Among them, Li et al. [3]in 2009, Sharma and Sharma [4] and Li et al. [5] in 2010, Zhou et al. [6] in 2011, Sharifi et al. [7]in 2012, Soleymani et al. [8], Soleymani and Babajee [9], Liu and Zhou [10] and Zhou et al. [11] in2013, Thukral [12] in 2014, Behl et al. [13] and Hueso et al. [14] in 2015, and Behl et al. [15] in 2016presented optimal fourth-order methods for multiple zeros. Additionally, Li et al. [5] (among otheroptimal methods) and Neta [16] presented non-optimal fourth-order iterative methods. In recentyears, efforts have been made to obtain an optimal scheme with a convergence order greater thanfour for multiple zeros with multiplicity m ≥ 1 of univariate function. Some of them only succeededin developing iterative schemes of a maximum of sixth-order convergence, in the case of multiplezeros; for example, see [17,18]. However, there are only few multipoint iterative schemes with optimaleighth-order convergence for multiple zeros which have been proposed very recently.



Behl et al. [19] proposed a family of optimal eighth-order iterative methods for multiple rootsinvolving univariate and bivariate weight functions given as:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

zn = yn − unQ(hn)f (xn)

f ′(xn), (2)

xn+1 = zn − untnG(hn, tn)f (xn)

f ′(xn),

where weight functions Q : C→ C and G : C2 → C are analytical in neighborhoods of (0) and (0, 0),

respectively, with un =(

f (yn)f (xn)

) 1m , hn = un

a1+a2unand tn =

(f (zn)f (yn)

) 1m , being a1 and a2 complex nonzero

free parameters.A second optimal eighth-order scheme involving parameters has been proposed by

Zafar et al. [20], which is given as follows:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

zn = yn −mun H(un)f (xn)

f ′(xn), (3)

xn+1 = zn − untn(B1 + B2un)P(tn)G(wn)f (xn)

f ′(xn),

where B1,B2 ∈ R are free parameters and the weight functions H : C→ C, P : C→ C and G : C→ C

are analytic in the neighborhood of 0 with un =(

f (yn)f (xn)

) 1m , tn =

(f (zn)f (yn)

) 1m and wn =

(f (zn)f (xn)

) 1m .

Recently, Geum et al. [21] presented another optimal eighth-order method for multiple roots:

yn = xn −mf (xn)

f ′(xn), n ≥ 0

wn = xn −mL f (s)f (xn)

f ′(xn), (4)

xn+1 = xn −m[

L f (s) + K f (s, u)] f (xn)

f ′(xn),

where L f : C → C is analytic in the neighborhood of 0 and K f : C2 → C is holomorphic in the

neighborhood of (0, 0) with s =(

f (yn)f (xn)

) 1m , u =

(f (wn)f (yn)

) 1m .

Behl et al. [22] also developed another optimal eighth-order method involving free parametersand a univariate weight function as follows:

yn = xn −mf (xn)

f ′(xn),

zn = yn −muf (xn)

f ′(xn)

1 + βu1 + (β− 2)u

, β ∈ R (5)

xn+1 = zn − uvf (xn)

f ′(xn)

(α1 + (1 + α2v)Pf (u)

),

where α1, α2 ∈ R are two free disposable parameters and the weight function Pf : C→ C is an analytic

function in a neighborhood of 0 with u =(

f (yn)f (xn)

) 1m , v =

(f (zn)f (yn)

) 1m .

109


Most recently, Behl at al. [23] presented an optimal eighth-order method involving univariateweight functions given as:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

zn = yn −munGf (un)f (xn)

f ′(xn), (6)

xn+1 = zn +unwn

1− wn

f (xn)

f ′(xn)

(Hf (un) + K f (vn)

),

where B1, B2 ∈ R are free parameters and the weight functions Gf , Hf , K f : C→ C are analytic in the

neighborhood of 0 with un =(

f (yn)f (xn)

) 1m , vn =

(f (zn)f (xn)

) 1m and wn =

(f (zn)f (yn)

) 1m .

Motivated by the research going on in this direction and with a need to give more stableoptimal higher-order methods, we propose a new family of optimal eighth-order iterative methods forfinding simple as well as multiple zeros of a univariate nonlinear function with multiplicity m ≥ 1.The derivation of the proposed class is based on a univariate and trivariate weight function approach.In addition, our proposed methods not only give the faster convergence but also have smaller residualerror. We have demonstrated the efficiency and robustness of the proposed methods by performingseveral applied science problems for numerical tests and observed that our methods have betternumerical results than those obtained by the existing methods. Further, the dynamical performance ofthese methods on the above mentioned problems supports the theoretical aspects, showing a goodbehavior in terms of dependence on initial estimations.

The rest of the paper is organized as follows: Section 2 provides the construction of thenew family of iterative methods and the analysis of convergence to prove the eighth order ofconvergence. In Section 3, some special cases of the new family are defined. In Section 4, the numericalperformance and comparison of some special cases of the new family with the existing ones are given.The numerical comparisons is carried out using the nonlinear equations that appear in the modeling ofthe predator–prey model, beam designing model, electric circuit modeling, and eigenvalue problem.Additionally, some dynamical planes are provided to compare their stability with that of knownmethods. Finally, some conclusions are stated in Section 5.

2. Construction of the Family

This section is devoted to the main contribution of this study, the design and convergenceanalysis of the proposed scheme. We consider the following optimal eighth-order class for findingmultiple zeros with multiplicity m ≥ 1:

yn = xn −mf (xn)

f ′(xn), n ≥ 0,

zn = yn −munG (un)f (xn)

f ′(xn),

xn+1 = zn −mun H(un, tn, wn)f (xn)

f ′(xn),

(7)

where G : C → C and H : C3 → C are analytical functions in a neighborhood of (0) and (0, 0, 0),

respectively being un =(

f (yn)f (xn)

) 1m , tn =

(f (zn)f (yn)

) 1m and wn =

(f (zn)f (xn)

) 1m .

In the next result, we demonstrate that the order of convergence of the proposed family reachesoptimal order eight.

110


Theorem 1. Let us consider x = ξ (say) is a zero with multiplicity m ≥ 1 of the involved function f .In addition, we assume that f : C → C is an analytical function in the region enclosing the multiple zero ξ.The proposed class defined by Equation (7) has an optimal eighth order of convergence, when the followingconditions are satisfied:

G(0) = 1, G1 = G′(0) = 2, G2 = G′′(0) = 4− G3

6, G3 = G

′′′(0)

H000 = 0, H100 = 0, H010 = 1, H101 = 3− G3

12, H110 = 2− H001,

H011 = 4, H020 = 1, |G3| < ∞, |H001| < ∞,

(8)

where Hijk =1

i!j!k!∂i+j+k

∂ujn∂tj

n∂wkn

H(un, tn, wn)|(un=0,tn=0,wn=0) for 0 ≤ i, k ≤ 1, 0 ≤ j ≤ 2.

Proof. Let us assume that en = xn − ξ is the error at nth step. By expanding f (xn) and f ′(xn) aboutx = ξ using Taylor series expansion, we have:

f (xn) =f (m)(ξ)

m!em

n

(1 + c1en + c2e2

n + c3e3n + c4e4

n + c5e5n + c6e6

n + c7e7n + c8e8

n + O(e9n))

(9)

and:

f ′(xn) =f (m)(ξ)

m!em−1

n

(m + c1(m + 1)en + c2(m + 2)e2

n + c3(m + 3)e3n + c4(m + 4)e4

n + c5(m + 5)e5n

+c6(m + 6)e6n + c7(m + 7)e7

n + c8(m + 8)e8n + O(e9

n))

,(10)

respectively, where ck =m!

(m + k)!f (m+k)(ξ)

f (m)(ξ), k = 1, 2, 3, . . . 8.

By inserting the above Equations (9) and (10), in the first substep of Equation (7), we obtain:

yn − ξ =c1e2

nm

+

(2mc2 − (m + 1)c2

1

)e3

n

m2 +4

∑k=0

Akek+4n + O(e9

n), (11)

where Ak = Ak(m, c1, c2, . . . , c8) are given in terms of m, c1, c2, c3, . . . , c8 with two explicitly

written coefficients A0 = 1m3

{3c3m2 + c3

1(m + 1)2 − c1c2m(3m + 4)}

and A1 = − 1m4

{c4

1(m + 1)3 −2c2c2

1m(2m2 + 5m + 3) + 2c3c1m2(2m + 3) + 2m2(c22(m + 2)− 2c4m

)}, etc.

With the help of Taylor series expansion and Equation (11), we get:

f (yn) = f (m)(ξ)e2mn

⎡⎢⎣ ( c1m )m

m!+

(2mc2 − (m + 1)c2

1

)( c1

m )men

m!c1+

6

∑k=0

Akek+2n + O(e9

n)

⎤⎥⎦ . (12)

Using Equations (9) and (12), we have:

un =

(f (yn)

f (xn)

) 1m=

c1en

m+

(2mc2 − (m + 2)c2

1

)e2

n

m2 + τ1e3n + τ2e4

n + τ3e5n + O(e6

n), (13)

where τ1 = 12m3

[c3

1(2m2 + 7m + 7) + 6c3m2 − 2c1c2m(3m + 7)], τ2 = − 1

6m4

[c4

1(6m3 + 29m2 +

51m + 34) − 6c2c21m(4m2 + 16m + 17) + 12c1c3m2(2m + 5) + 12m2(c2

2(m + 3) − 2c4m)]

and τ3 =

111


124m5

[− 24m3(c2c3(5m+ 17)− 5c5m

)+ 12c3c2

1m2(10m2 + 43m+ 49) + 12c1m2{

c22(10m2 + 47m+ 53)−

2c4m(5m + 13)}− 4c2c3

1m(30m3 + 163m2 + 306m + 209) + c51(24m4 + 146m3 + 355m2 + 418m + 209)

].

It is clear from Equation (13) that un is of order one. Therefore, we can expand the weight functionG(un) in the neighborhood of origin by Taylor series expansion up to third-order terms for eighth-orderconvergence as follows:

G(un) ≈ G(0) + unG′(0) + u2n

2!G′′(0) + u3

n3!

G′′′(0). (14)

Now, by inserting Equations (11)–(14) in the second substep of the proposed class (Equation (7)),we obtain:

zn − ξ = − (G(0)− 1)c1

me2

n −((1 + G′(0) + m− G(0)(m + 3))c2

1 + 2mc2(G(0)− 1))m2 e3

n

+5

∑j=1

Bjej+3n + O(e9

n),(15)

where Bj = Bj(G(0), G′(0), G′′(0), G′′′(0), m, c1, c2, . . . , c7), j = 1, 2, 3, 4, 5.In order to obtain fourth-order convergence, the coefficients of e2

n and e3n must be simultaneously

equal to zero. Thus, from Equation (15), we obtain the following values of G(0) and G′(0) :

G(0) = 1, G′(0) = 2. (16)

Using Equation (16), we have:

zn − ξ = − ((9− G′′(0) + m)c21 − 2mc1c2)

2m3 e4n +

4

∑j=1

Pjej+4n + O(e9

n), (17)

where Pj = Pj(G′′(0), G′′′(0), m, c1, c2, . . . , c7), j = 1, 2, 3, 4.With the help of Equation (17) and Taylor series expansion, we have:

f (zn) = f (m)(ξ)e4mn

⎡⎢⎢⎢⎢⎣2−m

((9−G′′(0)+m)c3

1−2mc1c2m3

)m

m!−

(2−m

((9−G′′(0)+m)c3

1−2mc1c2m3

)m−1η0

)3(m3m!)

en

+7

∑j=1

Pjej+1n + O(e9

n)

],

(18)

where:

η0 =(

124 + G′′′ (0)− 3G′′(0)(7 + 3m)c41 − 6m(−3G′′(0) + 4(7 + m)c2

1c2 + 12m2c22 + 12m2c1c3)

)and Pj = Pj(G′′(0), G′′′(0), m, c1, c2, . . . , c7), j = 1, 2, . . . 7.

Using Equations (9), (12) and (18), we further obtain:

tn =

(f (zn)

f (yn)

) 1m=

c21(9− G′′(0) + m)− 2mc2

2m2 e2n +

4

∑j=1

Qjej+2n + O(e7

n), (19)

112


where Qj = Qj(G′′(0), G′′′(0), m, c1, c2, . . . , c6), j = 1, 2, 3, 4 and:

wn =

(f (zn)

f (xn)

) 1m=

c31(9− G′′(0) + m)− 2mc1c2

2m3 e3n +

4

∑j=1

Qjej+3n + O(e8

n), (20)

where Qj = Qj(G′′(0), G′′′(0), m, c1, c2, . . . , c6), j = 1, 2, 3, 4.Hence, it is clear from Equation (13) that tn and wn are of order 2 and 3, respectively. Therefore, we

can expand weight function H(un, tn, wn) in the neighborhood of (0, 0, 0) by Taylor series expansionup to second-order terms as follows:

H(un, tn, wn) ≈ H000 + un H100 + tn H010 +wn H001 + untn H110 + unwn H101 +wntnH110 + t2nH020, (21)

where Hijk =1

i!j!k!∂i+j+k

∂ujn∂tj

n∂wkn

H(un, tn, wn)|(0,0,0), for 0 ≤ i, k ≤ 1, 0 ≤ j ≤ 2.

Using Equations (9)–(21) in the last substep of proposed scheme (Equation (7)), we have:

en+1 = −H000c1

me2

n +5

∑i=1

Eiei+2n + O(e8

n), (22)

where Ei = Ei(m, G′′(0), G′′′(0), H000, H100, H010, H001, H101, H110, H020, c1, c2, . . . , c6), i = 1, 2, 3, 4.From Equation (22), it is clear that we can easily obtain at least cubic order of convergence, for:

H000 = 0. (23)

Moreover, E1 = 0 for H100 = 0, we also have:

E2 =(−1 + H010) c1(−9 + G′′ (0) c2

1 + 2mc2)

2m3 .

Thus, we take:− 1 + H010 = 0, (24)

Thus, by inserting Equation (24), it results that E2 = 0 and:

E3 =(−2 + H001 + H110) c2

1(−9 + G′′ (0) c21 + 2mc2)

2m4 . (25)

Therefore, by taking:H110 = 2− H001, (26)

we have at least a sixth-order convergence. Additionally, for H020 = 1:

E4 =(−2 + 2H101 − G′′ (0)) c3

1(−9 + G′′ (0) c21 + 2mc2)

4m5 , (27)

which further yields:

H101 = 1 +G′′ (0)

2. (28)

Finally, we take:

H011 = 4, G′′ (0) = 4− G3

6.

where G3 = G′′′(0) .

113


Then, by substituting Equations (23), (24), (26) and (28) in Equation (22), we obtain the followingoptimal asymptotical error constant term:

en+1 =c1

288m7 (G3 + 6(5 + m))c21 − 12mc2)((G3(25 + m) + 2(227 + 90m + 7m2))c4

1

− 2m(180 + G3 + 24m)c21c2 + 24m2c2

2 + 24m2c1c3)e8n + O(e9

n).(29)

Equation (29) reveals that the proposed scheme (Equation (7)) reaches optimal eighth-orderconvergence using only four functional evaluations (i.e., f (xn), f ′(xn), f (yn) and f (zn)) per iteration.This completes the proof.

3. Some Special Cases of Weight Function

In this section, we discuss some special cases of our proposed class (7) by assigning differentkinds of weight functions. In this regard, please see the following cases, where we have mentionedsome different members of the proposed family.

Case 1: Let us describe the following polynomial weight functions directly from the hypothesis ofTheorem 1:

G(un) = 1 + 2un +

(2− G3

12

)u2

n +16

G3u3n,

H(un, tn, wn) = tn +

(H001 +

(3− G3

12

)un

)wn + ((2− H001) un + 4wn + tn) tn,

(30)

where H001 and G3 are free parameters.Case 1A: When H001 = 2, G3 = 0, we obtain the corresponding optimal eighth-order iterative

method as follows:

yn = xn −mf (xn)

f ′(xn), n ≥ 0,

zn = yn −mun(1 + 2un + 2u2n)

f (xn)

f ′(xn),

xn+1 = zn −mun(t2n + wn(2 + 3un + 4tn) + tn)

f (xn)

f ′(xn).

(31)

Case 2: Now, we suggest a mixture of rational and polynomial weight functions satisfyingcondition Equation (8) as follows:

G(un) =1 + a0un

1 + (a0 − 2) un + a3u2n

,


(H001 +

(3− G3

12

)un

)wn + ((2− H001) un + 4wn + tn) tn,

(32)

where a3 = −2 (a0 − 1) + G312 and a0, G3 and H001 are free parameters.

Case 2A: When a0 = 2, H001 = 2, G3 = 12, the corresponding optimal eighth-order iterativescheme is given by:

yn = xn −mf (xn)

f ′(xn), n ≥ 0,

zn = yn −mun

(1 + 2un

1− u2n

)f (xn)

f ′(xn),

xn+1 = zn −mun(tn + 2 (1 + un)wn + (tn + 4wn) tn)f (xn)

f ′(xn).

(33)

114


Case 3: Now, we suggest another rational and polynomial weight function satisfying Equation (8)as follows:

G(un) =1 + a0un

1 + (a0 − 2) un + a3u2n + a4u3

n,


(H001 +

(3− G3

12

)un

)wn + ((2− H001) un + 4wn + tn) tn,

(34)

where a3 = −2(a0 − 1) + G312 , a4 = 2a0 + (a0 − 6) G3

12 and a0, H001 and G3 are free.Case 3A: By choosing a0 = 4, a3 = −5, a4 = 6, H001 = 2, G3 = 12, the corresponding optimal

eighth-order iterative scheme is given by:

yn = xn −mf (xn)

f ′(xn), n ≥ 0,

zn = yn −mun

(1 + 4un

1 + 2un − 5u2n + 6u3

n

)f (xn)

f ′(xn)

xn+1 = zn −mun(tn + 2 (1 + un)wn + (tn + 4wn) tn)f (xn)

f ′(xn).

(35)

In a similar way, we can develop several new and interesting optimal schemes with eighth-orderconvergence for multiple zeros by considering new weight functions which satisfy the conditions ofTheorem 1.

4. Numerical Experiments

This section is devoted to demonstrating the efficiency, effectiveness, and convergence behavior ofthe presented family. In this regard, we consider some of the special cases of the proposed class,namely, Equations (31), (33) and (35), denoted by NS1, NS2, and NS3, respectively. In addition, wechoose a total number of four test problems for comparison: The first is a predator–prey model, thesecond is a beam designing problem, the third is an electric circuit modeling for simple zeros, and thelast is an eigenvalue problem.

Now, we want to compare our methods with other existing robust schemes of the same order onthe basis of the difference between two consecutive iterations, the residual errors in the function,the computational order of convergence ρ, and asymptotic error constant η. We have choseneighth-order iterative methods for multiple zeros given by Behl et al. [19,23]. We take the followingparticular case (Equation (27)) for (a1 = 1, a2 = −2, G02 = 2m) of the family by Behl et al. [19] anddenote it by BM1 as follows:

yn = xn −mf (xn)

f ′(xn),

zn = yn −m (1 + 2hn)f (xn)

f ′(xn)un, (36)

xn+1 = zn −m(

1 + tn + t2n + 3h2

n + hn(2 + 4tn − 2hn)) f (xn)

f ′(xn)untn.

115


From the eighth-order family of Behl et al. [23], we consider the following special case denotedby BM2:

yn = xn −mf (xn)

f ′(xn),

zn = yn −mun (1 + 2un)f (xn)

f ′(xn), (37)

xn+1 = zn + munwn

1− wn

f (xn)

f ′(xn)

(−1− 2un − u2

n + 4u3n − 2vn

).

Tables 1–4 display the number of iteration indices (n), the error in the consecutive iterations|xn+1 − xn|, the computational order of convergence ρ ≈ log| f (xn+1)/ f (xn)|

log|( f (xn)/ f (xn−1))| , n ≥ 1, (the formula byJay [24]), the absolute residual error of the corresponding function (| f (xn)|), and the asymptotical error

constant η ≈∣∣∣∣ en

e8n−1

∣∣∣∣. We did our calculations with 1000 significant digits to minimize the round-off

error. We display all the numerical values in Tables 1–4 up to 7 significant digits with exponent. Finally,we display the values of approximated zeros up to 30 significant digits in Examples 1–4, although aminimum of 1000 significant digits are available with us.

For computer programming, all computations have been performed using the programmingpackage Maple 16 with multiple precision arithmetics. Further, the meaning of a(±b) is a× 10(±b) inTables 1–4.

Now, we explain the real life problems chosen for the sake of comparing the schemes as follows:

Example 1 (Predator-Prey Model). Let us consider a predator-prey model with ladybugs as predators andaphids as preys [25]. Let x be the number of aphids eaten by a ladybug per unit time per unit area, called thepredation rate, denoted by P(x). The predation rate usually depends on prey density and is given as:

P (x) = Kx3

a3 + x3 , a, K > 0.

Let the growth of aphids obey the Malthusian model; therefore, the growth rate of aphids G per hour is:

G (x) = rx, r > 0.

The problem is to find the aphid density x for which:

P (x) = G (x) .

This gives:rx3 − Kx2 + ra3 = 0.

Let K = 30 aphids eaten per hour, a = 20 aphids and r = 2− 13 per hour. Thus, we are required to find the

zero of:f1 (x) = 0.7937005260x3 − 30x2 + 6349.604208

The desired zero of f1 is 25.198420997897463295344212145564 with m = 2. We choose x0 = 20.

116


Table 1. Comparison of different multiple root finding methods for f1(x).

BM1 BM2 NS1 NS2 NS3

|x1 − x0| 2.064550(1) 4.789445 1.219414(1) 1.214342(1) 1.213887(1)| f (x1)| 1.008384(4) 4.963523 1.739946(3) 1.712863(3) 1.710446(3)|x2 − x1| 1.544682(1) 4.088744(−1) 6.995715 6.944984 6.940438| f (x2)| 1.967429(−6) 3.035927(−7) 3.672323(−9) 6.792230(−9) 4.951247(−9)|x3 − x2| 2.560869(−4) 1.005971(−4) 1.106393(−5) 1.504684(−5) 1.284684(−5)| f (x3)| 5.685107(−81) 6.093227(−29) 1.223217(−100) 5.427728(−98) 2.522949(−99)

η 7.900841 0.1287852 1.928645(−12) 2.780193(−12) 2.386168(−12)ρ 7.676751 3.0078946 7.834927 7.814388 7.825421

Example 2 (Beam Designing Model). We consider a beam positioning problem (see [26]) where an r meterlong beam is leaning against the edge of the cubical box with sides of length 1 m each, such that one of its endstouches the wall and the other touches the floor, as shown in Figure 1.

Figure 1. Beam positioning problem.

What should be the distance along the floor from the base of the wall to the bottom of the beam? Let y be thedistance in meters along the beam from the floor to the edge of the box and let x be the distance in meters from thebottom of the box to the bottom of the beam. Then, for a given value of r, we have:

f2 (x) = x4 + 4x3 − 24x2 + 16x + 16 = 0.

The positive solution of the equation is a double root x = 2. We consider the initial guess x0 = 1.7.


BM1 BM2 NS1 NS2 NS3

|x1 − x0| 1.288477 2.734437(−1) 7.427026(−1) 7.391615(−1) 7.388023(−1)| f (x1)| 35.99479 1.670143(−2) 5.783224 5.682280 5.672098|x2 − x1| 9.884394(−1) 2.654643(−2) 4.427007(−1) 4.391589(−1) 4.388001(−1)| f (x2)| 3.566062(−8) 2.333107(−9) 8.652078(−11) 1.664205(−10) 1.1624462(−10)|x3 − x2| 3.854647(−5) 9.859679(−6) 1.898691(−6) 2.633282(−7) 2.200800(−6)| f (x3)| 7.225712(−77) 5.512446(−30) 2.306147(−95) 1.620443(−92) 4.8729521(−94)

e η 4.230427(−1) 3.997726(7) 1.286982(−3) 1.903372(−3) 1.601202(−3)ρ 7.629155 3.0090640 7.812826 7.7859217 7.800775

Example 3 (The Shockley Diode Equation and Electric Circuit). Let us consider an electric circuitconsisting of a diode and a resistor. By Kirchoff’s voltage law, the source voltage drop VS is equal to thesum of the voltage drops across the diode VD and resistor VR :

Vs = VR + VD. (38)

117


Let the source voltage be VS = 0.5 V and from Ohm’s law:

VR = RI. (39)

Additionally, the voltage drop across the diode is given by the Shockley diode equation as follows:

I = IS

(e

VDnVT − 1

), (40)

where I is the diode current in amperes, IS is saturation current (amperes), n is the emission or ideality constant(1 ≤ n ≤ 2 for silicon diode), and VD is the voltage applied across the diode. Solving Equation (40) for VD andusing all the values in Equation (38), we obtain:

−0.5 + RI + nVT ln(

IIS

+ 1)= 0

Now, for the given values of n, VT, R and IS, we have the following equation [27]:

−0.5 + 0.1I + 1.4 ln (I + 1) = 0.

Replacing I with x, we have

f3 (x) = −0.5 + 0.1x + 1.4 ln (x + 1) .

The true root of the equation is 0.389977198390077586586453532646. We take x0 = 0.5.


BM1 BM2 NS1 NS2 NS3

|x1 − x0| 1.100228(−1) 1.100228(−1) 1.100228(−1) 1.100228(−1) 1.100228(−1)| f (x1)| 3.213611(−12) 1.902432(−10) 7.591378(−11) 4.728795(−10) 1.626799(−10)|x2 − x1| 2.902439(−12) 1.718220(−10) 6.856308(−11) 4.270907(−10) 1.469276(−10)| f (x2)| 9.512092(−97) 6.797214(−81) 2.215753(−84) 2.393956(−77) 1.758525(−81)|x3 − x2| 8.591040(−97) 6.139043(−81) 2.001202(−84) 2.162151(−77) 1.588247(−81)| f (x3)| 5.604505(−773) 1.805114(−644) 1.1671510(−672) 1.032863(−615) 3.278426(−649)

η 1.705849(−4) 8.081072(−3) 4.097965(−3) 1.953099(−2) 7.312887(−3)ρ 7.999999 7.999999 7.999999 7.999999 7.999999

Example 4 (Eigenvalue Problem). One of the challenging task of linear algebra is to calculate the eigenvalues ofa large square matrix, especially when the required eigenvalues are the zeros of the characteristic polynomialobtained from the determinant of a square matrix of order greater than 4. Let us consider the following9 × 9 matrix:

A =18

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−12 0 0 19 −19 76 −19 18 437−64 24 0 −24 24 64 −8 32 376−16 0 24 4 −4 16 −4 8 92−40 0 0 −10 50 40 2 20 242−4 0 0 −1 41 4 1 2 25−40 0 0 18 −18 104 −18 20 462−84 0 0 −29 29 84 21 42 50116 0 0 −4 4 −16 4 16 −920 0 0 0 0 0 0 0 24

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The corresponding characteristic polynomial of matrix A is given as follows:

f4(x) = x9 − 29x8 + 349x7 − 2261x6 + 8455x5 − 17663x4 + 15927x3 + 6993x2 − 24732x + 12960. (41)

118


The above function has one multiple zero at ξ = 3 of multiplicity 4 with initial approximation x0 = 3.1.


BM1 BM2 NS1 NS2 NS3

|x1 − x0| 1.577283(−1) 9.9275251(−2) 1283418(−1) 1.283182(−1) 1.283180(−1)| f (x1)| 9.361198(−4) 2.205656(−11) 5.299339(−5) 5.281568(−5) 5.281425(−5)|x2 − x1| 5.772837(−2) 7.247474(−4) 2.834188(−2) 2.831824(−2) 2.831805(−2)| f (x2)| 9.059481(−49) 7.0590148(−38) 2.755794(−55) 8.779457(−55) 5.772523(−55)|x3 − x2| 3.262145(−13) 2.278878(−10) 7.661066(−15) 1.023515(−14) 9.216561(−15)| f (x3)| 4.543117(−408) 2.278878(−117) 4.807225(−457) 1.869778(−452) 4.077620(−454)

η 2.644775(−3) 2.264227(15) 1.840177(−2) 2.474935(−2) 2.228752(−2)ρ 7.981915 3.000250 7.989789 7.988696 7.989189

In Tables 1–4, we show the numerical results obtained by applying the different methods forapproximating the multiple roots of f1(x)− f4(x). The obtained values confirm the theoretical results.From the tables, it can be observed that our proposed schemes NS1, NS2, and NS3 exhibit a betterperformance in approximating the multiple root of f1, f2 and f4 among other similar methods. Only inthe case of the example for simple zeros Behl’s scheme BM1 is performing slightly better than the othermethods.

Dynamical Planes

The dynamical behavior of the test functions is presented in Figures 2–9. The dynamical planeshave been generated using the routines published in Reference [28]. We used a mesh of 400× 400points in the region of the complex plane [−100, 100]× [−100, 100]. We painted in orange the pointswhose orbit converged to the multiple root and in black those points whose orbit either diverged orconverged to a strange fixed point or a cycle. We worked out with a tolerance of 10−3 and a maximumnumber of 80 iterations. The multiple root is represented in the different figures by a white star.

20 21 22 23 24 25 26

Re{z}

-3

-2

-1

0

1

2

3

Im{z

}

20 21 22 23 24 25 26

Re{z}

-3

-2

-1

0

1

2

3

Im{z

}

20 21 22 23 24 25 26

Re{z}

-3

-2

-1

0

1

2

3

Im{z

}

Figure 2. Dynamical planes of the methods NS1 (Left), NS2 (Center), and NS3 (Right) for f1(x).

20 21 22 23 24 25 26

Re{z}

-3

-2

-1

0

1

2

3

Im{z

}

20 21 22 23 24 25 26

Re{z}

-3

-2

-1

0

1

2

3

Im{z

}

Figure 3. Dynamical planes of the methods BM1 (Left) and BM2 (Right) for f1(x).

119


1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}


1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

Figure 5. Dynamical planes of the methods BM1 (Left) and BM2 (Right) on f2(x).

-2 -1 0 1 2

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

-2 -1 0 1 2

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

-2 -1 0 1 2

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}


-2 -1 0 1 2

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

-2 -1 0 1 2

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}


120


1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

Figure 8. Dynamical planes of the methods NS1 (Left), NS2 (Center), and NS3 (Right) on f4(x).

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}

1 2 3 4 5

Re{z}

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Im{z

}


Figures 2–9 study the convergence and divergence regions of the new schemes NS1, NS2, and NS3in comparison with the other schemes of the same order. In the case of f1(x) and f2(x), we observedthat the new schemes are more stable than BM1 and BM2 as they are almost divergence-free andalso converge faster than BM1 and BM2 in their common regions of convergence. In the case off3(x), BM1 performs better; however, NS1, NS2, and NS3 have an edge over BM2 for the regionin spite of the analogous behavior to BM2, as the new schemes show more robustness. Similarly,in the case of f4(x), it can be clearly observed that the divergence region for BM1 is bigger than that forNS1, NS2, and NS3. Additionally, these schemes perform better than BM2 where they are convergent.The same behavior can be observed through the numerical comparison of these methods in Tables 1–4.As a future extension, we shall be trying to construct a new optimal eighth-order method whosestability analysis can allow to choose the optimal weight function for the best possible results.

5. Conclusions

In this manuscript, a new general class of optimal eighth-order methods for solving nonlinearequations with multiple roots was presented. This family was obtained using the procedure of weightfunctions with two functions: One univariate and another depending on three variables. To reachthis optimal order, some conditions on the functions and their derivatives must be imposed. Severalspecial cases were selected and applied to different real problems, comparing their performance withthat of other known methods of the same order of convergence. Finally, their dependence on initialestimations was analyzed from their basins of attraction.

Author Contributions: methodology, F.Z.; writing original draft preparation, F.Z.; writing review and editing,J.R.T.; visualization, A.C.; supervision, J.R.T.

Funding: This research was partially supported byMinisterio de Economía y CompetitividadMTM2014-52016-C2-2-P, by Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Facultyfor Future Program.


121


References

1. Schroder, E. Uber unendlich viele Algorithmen zur Auflosung der Gleichungen. Math. Ann. 1870, 2, 317–365.[CrossRef]

2. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 1974, 21,643–651. [CrossRef]

3. Li, S.; Liao, X.; Cheng, L. A new fourth-order iterative method for finding multiple roots of nonlinearequations. Appl. Math. Comput. 2009, 215, 1288–1292.

4. Sharma, J.R.; Sharma, R. Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 2010,217, 878–881. [CrossRef]

5. Li, S.G.; Cheng, L.Z.; Neta, B. Some fourth-order nonlinear solvers with closed formulae for multiple roots.Comput. Math. Appl. 2010, 59, 126–135. [CrossRef]

6. Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the muliplte roots of nonlinearequations. J. Comput. Math. Appl. 2011, 235, 4199–4206. [CrossRef]

7. Sharifi, M.; Babajee, D.K.R.; Soleymani, F. Finding the solution of nonlinear equations by a class of optimalmethods. Comput. Math. Appl. 2012, 63, 764–774. [CrossRef]

8. Soleymani, F.; Babajee, D.K.R.; Lofti, T. On a numerical technique forfinding multiple zeros and its dynamic.J. Egypt. Math. Soc. 2013, 21, 346–353. [CrossRef]

9. Soleymani, F.; Babajee, D.K.R. Computing multiple zeros using a class of quartically convergent methods.Alex. Eng. J. 2013, 52, 531–541. [CrossRef]

10. Liu, B.; Zhou, X. A new family of fourth-order methods for multiple roots of nonlinear equations. Non. Anal.Model. Cont. 2013, 18, 143–152.

11. Zhou, X.; Chen, X.; Song, Y. Families of third and fourth order methods for multiple roots of nonlinearequations. Appl. Math. Comput. 2013, 219, 6030–6038. [CrossRef]

12. Thukral, R. A new family of fourth-order iterative methods for solving nonlinear equations with multipleroots. J. Numer. Math. Stoch. 2014, 6, 37–44.

13. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. On developing fourth-order optimal families of methodsfor multiple roots and their dynamics. Appl. Math. Comput. 2015, 265, 520–532. [CrossRef]

14. Hueso, J.L.; Martínez, E.; Teruel, C. Determination of multiple roots of nonlinear equations and applications.J. Math. Chem. 2015, 53, 880–892. [CrossRef]

15. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R.; Kanwar, V. An optimal fourth-order family of methods formultiple roots and its dynamics. Numer. Algor. 2016, 71, 775–796. [CrossRef]

16. Neta, B. Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 2010,87, 1023–1031. [CrossRef]

17. Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modifieddouble-Newton type and their dynamics. Appl. Math. Comput. 2015, 270, 387–400 . [CrossRef]

18. Geum, Y.H.; Kim, Y.I.; Neta, B. A sixth-order family of three-point modified Newton-like multiple-rootfinders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 2016, 283, 120–140.[CrossRef]

19. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. An eighth-order family of optimal multiple root findersand its dynamics. Numer. Algor. 2017. [CrossRef]

20. Zafar, F.; Cordero, A.; Rana, Q.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots ofnonlinear equations using free parameters. J. Math. Chem. 2017. [CrossRef]

21. Geum, Y.H.; Kim, Y.I.; Neta, B. Constructing a family of optimal eighth-order modified Newton-typemultiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points.J. Comput. Appl. Math. 2018, 333, 131–156. [CrossRef]

22. Behl, R.; Zafar, F.; Alshomrani, A.S.; Junjua, M.; Yasmin, N. An optimal eighth-order scheme for multiplezeros of univariate function. Int. J. Comput. Math. 2018, 15. [CrossRef]

23. Behl, R.; Alshomrani, A.S.; Motsa, S.S. An optimal scheme for multiple roots of nonlinear equations witheighth-order convergence. J. Math. Chem. 2018. [CrossRef]

24. Jay, L.O. A note on Q-order of convergence. BIT Numer. Math. 2001, 41, 422–429. [CrossRef]25. Edelstein-Keshet, L. Differential Calculus for the Life Sciences; Univeristy of British Columbia: Vancouver, BC,

Canada, 2017.

122


26. Zachary, J.L. Introduction to Scientific Programming: Computational Problem Solving Using Maple and C; Springer:New York, NY, USA, 2012.

27. Khoury, R. Douglas Wilhelm Harder, Numerical Methods and Modelling for Engineering; Springer InternationalPublishing: Berlin, Germany, 2017.

28. Chicharro, F.I.; Cordero, A.; Torregrosa, J.R. Drawing dynamical and parameters planes of iterative familiesand methods. Sci. World J. 2013, 2013, 780153. [CrossRef] [PubMed]


123

mathematics

Article

A Higher Order Chebyshev-Halley-Type Family ofIterative Methods for Multiple Roots

Ramandeep Behl 1, Eulalia Martínez 2,*, Fabricio Cevallos 3 and Diego Alarcón 4

1 Department of Mathematics, King Abdualziz University, Jeddah 21589, Saudi Arabia;[email protected]

2 Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València,Camino de Vera s/n, 46022 Valencia, Spain

3 Fac. de Ciencias Económicas, Universidad Laica “Eloy Alfaro de Manabí”, Manabí 130214, Ecuador;[email protected]

4 Departamento de Matemática Aplicada, Universitat Politècnica de València, Camino de Vera s/n,46022 Valencia, Spain; [email protected]


Received: 15 January 2019; Accepted: 22 March 2019; Published: 9 April 2019

Abstract: The aim of this paper is to introduce new high order iterative methods for multiple roots ofthe nonlinear scalar equation; this is a demanding task in the area of computational mathematics andnumerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function havingat least sixth-order convergence and eighth-order convergence for a particular value in the case ofmultiple roots. With regard to computational cost, each member of our scheme needs four functionalevaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2,which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtainthe theoretical convergence order by using Taylor developments. Finally, we consider some real-lifesituations for establishing some numerical experiments to corroborate the theoretical results.

Keywords: nonlinear equations; multiple roots; Chebyshev–Halley-type; optimal iterative methods;efficiency index

1. Introduction

One important field in the area of computational methods and numerical analysis is to findapproximations to the solutions of nonlinear equations of the form:

f (x) = 0, (1)

where f : D ⊂ C→ C is the analytic function in the enclosed region D, enclosing the required solution.It is almost impossible to obtain the exact solution in an analytic way for such problems. Therefore,we concentrate on obtaining approximations of the solution up to any specific degree of accuracy bymeans of an iterative procedure, of course doing it also with the maximum efficiency. In [1], Kung andTraub conjectured that a method without memory that uses n + 1 functional evaluations per iterationcan have at most convergence order p = 2n. If this bound is reached, the method is said to be optimal.

For solving nonlinear Equation (1) by means of iterations, we have the well-knowncubically-convergent family of Chebyshev–Halley methods [2], which is given by:

xn+1 = xn −[

1 +12

L f (xn)

1− αL f (xn)

]f (xn)

f ′(xn), α ∈ R, (2)



where L f (xn) =f ′′(xn) f (xn){ f ′(xn)}2 . A great variety of iterative methods can be reported in particular cases.

For example, the classical Chebyshev’s method [1,3], Halley’s method [1,3], and the super-Halleymethod [1,3] can be obtained if α = 0, α = 1

2 , and α = 1, respectively. Despite the third-orderconvergence, the scheme (2) is considered less practical from a computational point of view because ofthe computation of the second-order derivative.

For this reason, several variants of Chebyshev–Halley’s methods free from the second-orderderivative have been presented in [4–7]. It has been shown that these methods are comparable tothe classical third-order methods of the Chebyshev–Halley-type in their performance and can alsocompete with Newton’s method. One family of these methods is given as follows:

yn = xn − f (xn)

f ′(xn),

xn+1 = xn −(

1 +f (yn)

f (xn)− α f (yn)

)f (xn)

f ′(xn), α ∈ R

(3)

We can easily obtain some well-known third-order methods proposed by Potra and Pták [4]and Sharma [5] (the Newton-secant method (NSM)) for α = 0 and α = 1. In addition, we haveOstrowski’s method [8] having optimal fourth-order convergence, which is also a special case forα = 2. This family is important and interesting not only because of not using a second- or higher orderderivative. However, this scheme also converges at least cubically and has better results in comparisonto the existing ones. Moreover, we have several higher order modifications of the Chebyshev–Halleymethods available in the literature, and some of them can be seen in [9–12].

In this study, we focus on the case of the multiple roots of nonlinear equations. We have somefourth-order optimal and non-optimal modifications or improvements of Newton’s iteration functionfor multiple roots in the research articles [13–17]. Furthermore, we can find some higher order methodsfor this case, but some of them do not reach maximum efficiency [18–23]; so, this topic is of interest inthe current literature.

We propose a new Chebyshev–Halley-type iteration function for multiple roots, which reaches ahigh order of convergence. Specifically, we get a family of iterative methods with a free parameter α,with sixth-order convergence. Therefore, the efficiency index is 61/4, and for α = 2, this index is 81/4,which is the maximum value that one can get with four functional evaluations, reaching optimality inthe sense of Kung and Traub’s conjecture. Additionally, an extensive analysis of the convergence orderis presented in the main theorem.

We recall that ξ ∈ C is a multiple root of the equation f (x) = 0, if it is verified that:

f (ξ) = 0, f ′(ξ) = 0, · · · , f (m−1)(ξ) = 0 and f (m)(ξ) �= 0,

the positive integer (m ≥ 1) being the multiplicity of the root.We deal with iterative methods in which the multiplicity must be known in advance, because

this value, m, is used in the iterative expression. However, we point out that these methods also workwhen one uses an estimation of the multiplicity, as was proposed in the classical study carried outin [24].

Finally, we consider some real-life situations that start from some given conditions to investigateand some standard academic test problems for numerical experiments. Our iteration functions hereare found to be more comparable and effective than the existing methods for multiple roots in termsof residual errors and errors among two consecutive iterations, and also, we obtain a more stablecomputational order of convergence. That is, the proposed methods are competitive.

125


2. Construction of the Higher Order Scheme

In this section, we present the new Chebyshev–Halley-type methods for multiple roots ofnonlinear equations, for the first time. In order to construct the new scheme, we consider thefollowing scheme:

yn = xn −mf (xn)

f ′(xn),

zn = xn −m(

1 +η

1− αη

)f (xn)

f ′(xn),

xn+1 = zn − H(η, τ)f (xn)

f ′(xn),

(4)

where the function:

H(η, τ) =ητ(

β− (α− 2)2η2(η + 1) + τ3 + τ2)(η + 1)(τ + 1)

with:

η =

(f (yn)

f (xn)

) 1m ,

τ =

(f (zn)

f (yn)

) 1m ,

β = m((α(α + 2) + 9)η3 + η2(α(α + 3)− 6τ − 3) + η(α + 8τ + 1) + 2τ + 1

),

where α ∈ R is a free disposable variable. For m = 1, we can easily obtain the scheme (3) from the firsttwo steps of the scheme (4).

In Theorem 1, we illustrate that the constructed scheme attains at least sixth-order convergenceand for α = 2, it goes to eighth-order without using any extra functional evaluation. It is interesting toobserve that H(η, τ) plays a significant role in the construction of the presented scheme (for details,please see Theorem 1).

Theorem 1. Let us consider x = ξ to be a multiple zero with multiplicity m ≥ 1 of an analytic function f : C→C in the region containing the multiple zero ξ of f (x). Then, the present scheme (4) attains at least sixth-orderconvergence for each α, but for a particular value of α = 2, it reaches the optimal eighth-order convergence.

Proof. We expand the functions f (xn) and f ′(xn) about x = ξ with the help of a Taylor’s seriesexpansion, which leads us to:

f (xn) =f (m)(ξ)

m!em

n

(1 + c1en + c2e2

n + c3e3n + c4e4

n + c5e5n + c6e6

n + c7e7n + c8e8

n + O(e9n)

), (5)

and:

f ′(xn) =f m(ξ)

m!em−1

n

(m + (m + 1)c1en + (m + 2)c2e2

n + (m + 3)c3e3n + (m + 4)c4e4

n + (m + 5)c5e5n

+ (m + 6)c6e6n + (m + 7)c7e7

n + (m + 8)c8e8n + O(e9

n)

),

(6)

respectively, where ck = m!(m−1+k)!

f m−1+k(ξ)f m(ξ)

, k = 2, 3, 4 . . . , 8 and en = xn − ξ is the error in the

nth iteration.Inserting the above expressions (5) and (6) into the first substep of scheme (4) yields:

yn − ξ =c1

me2

n +5

∑i=0

φiei+3n + O(e9

n), (7)

126


where φi = φi(m, c1, c2, . . . , c8) are given in terms of m, c2, c3, . . . , c8, for example φ0 = 1m2

(2mc2 − (m +

1)c21)

and φ1 = 1m3

[3m2c3 + (m + 1)2c3

1 −m(3m + 4)c1c2

], etc.

Using the Taylor series expansion and the expression (7), we have:

f (yn) = f (m)(ξ)e2mn

[( c1m) m

m!+

(2mc2 − (m + 1)c21)( c1

m)m en

m!c1+( c1

m

)1+m 12m!c3

1

{(3 + 3m + 3m2 + m3)c4

1

− 2m(2 + 3m + 2m2)c21c2 + 4(m− 1)m2c2

2 + 6m2c1c3}

e2n +

5

∑i=0

φiei+3n + O(e9

n)

].

(8)

We obtain the following expression by using (5) and (8):

η =c1en

m+

2mc2 − (m + 2)c21

m2 e2n + θ0e3

n + θ1e4n + θ2e5

n + O(e6n), (9)

where θ0 =(2m2+7m+7)c3

1+6m2c3−2m(3m+7)c1c22m3 , θ1 = − 1

6m4

[12m2(2m + 5)c1c3 + 12m2((m + 3)c2

2 −2mc4)− 6m(4m2 + 16m + 17)c2

1c2 + (6m3 + 29m2 + 51m + 34)c41

]and θ2 =

124m5

[12m2(10m2 + 43m +

49)c21c3− 24m3((5m+ 17)c2c3− 5mc5)+ 12m2

((10m2 + 47m+ 53)c2

2− 2m(5m+ 13)c4

)c1− 4m(30m3 +

163m2 + 306m + 209)c31c2 + (24m4 + 146m3 + 355m2 + 418m + 209)c5

1

].

With the help of Expressions (5)–(9), we obtain:

zn − ξ = − (α− 2)c21

m2 e3n +

4

∑i=0

ψiei+4n + O(e9

n), (10)

where ψi = ψi(α, m, c1, c2, . . . , c8) are given in terms of α, m, c2, c3, . . . , c8 with the first two coefficients

explicitly written as ψ0 = − 12m3

[(2α2− 10α + (7− 4α)m + 11

)c3

1 + 2m(4α− 7)c1c2

]and ψ1 = 1

6m4

[(−6α3 + 42α2− 96α+ (29− 18α)m2 + 6(3α2− 14α+ 14)m+ 67

)c4

1 + 12m2(5− 3α)c1c3 + 12m2(3− 2α)c22 +

12m(− 3α2 + 14α + (5α− 8)m− 14

)c2

1c2

].

By using the Taylor series expansion and (10), we have:

f (zn) = f (m)(ξ)e3mn

⎡⎢⎢⎣(− (α−2)c2

1m2

)m

m!+

5

∑i=1

ψiein + O(e6

n)

⎤⎥⎥⎦ . (11)

From Expressions (8) and (11), we further have:

τ =− (α− 2)c1

men +

((− 2α2 + 8α + (2α− 3)m− 7)c2

1 + 2m(3− 2α)c2)

2m2 e2n + γ1e3

n + γ2e4n + O(e5

n), (12)

where γ1 = 13m3

[(− 3α3 + 18α2 − 30α + (4− 3α)m2 + 3(2α2 − 7α + 5)m + 11)c3

1 + 3m2(4− 3α)c3 +

3m(−4α2 + 14α + 3αm− 4m− 10)c1c2

]and γ2 = 1

24m4

[24m2(− 6α2 + 20α + (4α− 5)m− 14

)c1c3 +

12m2((−8α2 + 24α + 4αm− 5m− 13)c22 + 2m(5− 4α)c4

)− 12m(12α3 − 66α2 + 100α + 2(4α− 5)m2 +

(−20α2 + 64α− 41)m− 33)c2

1c2 +(− 24α4 + 192α3 − 492α2 + 392α + 6(4α− 5)m3 + (−72α2 + 232α−

151)m2 + 6(12α3 − 66α2 + 100α− 33)m + 19)c4

1

].

By using Expressions (9) and (12), we obtain:

H(η, τ) = − (α− 2)c21

m2 e2n + λ1e3

n + λ2e4n + O(e5

n) (13)

127


where λ1 = c12m3

[c2

1(−2α2 + 8α + (4α− 7)m− 7

)+ 2(7 − 4α)c2m

]and λ2 = 1

6m3

[c4

1( − 6α3 +

36α2 − 66α + (29 − 18α)m2 + 3(6α2 − 22α + 17)m + 34)+ 12(5 − 3α)c3c1m2 + 12(3 − 2α)c2

2m2 +

6c2c21m

(−6α2 + 22α + 2(5α− 8)m− 17) ]

.Now, we use the expressions (5)–(13) in the last substep of Scheme (4), and we get:

en+1 =3

∑i=1

Liei+5n + O(e9

n), (14)

where L1 =(α−2)2c3

1m6

[c2

1(α2 − α + m2 − (α2 + 4α− 17

)m− 3

) − 2c2(m − 1)m], L2 = (α − 2)c2

1[ −

12c2c21m{

10α3 − 24α2 − 39α + (16α− 27)m2 − (10α3 + 27α2 − 262α + 301)m + 91}+ 12c3c1m2(−4α +

(4α − 7)m + 8) + 12c22m2(−12α + 4(3α − 5)m + 21) + c4

1{ − 24α4 + 168α3 − 156α2 − 662α + (52α −

88)m3 − (60α3 + 162α2 − 1616α + 1885)m2 + 2(18α4 − 12α3 − 711α2 + 2539α− 2089)m + 979}]

and

L3 = c124m8

[− 24c2c3c1m3((42α2 − 146α + 125)m − 6(7α2 − 26α + 24)

) − 24c32m3( − 24α2 + 84α +

(24α2 − 80α + 66)m− 73)+ 12c3c3

1m2{2(15α4 − 63α3 − 5α2 + 290α− 296) + (54α2 − 190α + 165)m2 +

(−30α4 − 28α3 + 968α2 − 2432α + 1697)m}+ 12c2

1m2{

c22

(80α4 − 304α3 − 226α2 + 1920α + 2(81α2 −

277α + 234)m2 + (−80α4 − 112α3 + 2712α2 − 6410α + 4209)m − 1787)− 4(α − 2)c4m(−3α + (3α −

5)m + 6)}− 2c2c4

1m{− 3(96α5 − 804α4 + 1504α3 + 2676α2 − 10612α + 8283) + 4(177α2 − 611α +

521)m3 − 3(220α4 + 280α3 − 7556α2 + 18400α− 12463)m2 + 4(108α5 − 234α4 − 4302α3 + 22902α2 −38593α + 20488)m

}+ c6

1

{48α6 − 480α5 + 996α4 + 5472α3 − 29810α2 + 50792α + (276α2 − 956α +

818)m4 + (−360α4 − 448α3 + 12434α2 − 30518α + 20837)m3 + (432α5 − 1236α4 − 16044α3 + 92306α2 −161292α + 88497)m2 + (−168α6 + 888α5 + 5352α4 − 55580α3 + 173290α2 − 224554α + 97939)m −29771

}].

It is noteworthy that we reached at least sixth-order convergence for all α. In addition, we caneasily obtain L1 = L2 = 0 by using α = 2.

Now, by adopting α = 2 in Expression (14), we obtain:

en+1 =A0(12c3c1m3 − 12c2c2

1m(3m2 + 30m− 1) + 12c22m2(2m− 1) + c4

1(10m3 + 183m2 + 650m− 3))

24m8 e8n + O(e9

n), (15)

where A0 = (c31(m + 1) − 2c1c2m). The above Expression (15) demonstrates that our proposed

Scheme (4) reaches eighth-order convergence for α = 2 by using only four functional evaluationsper full iteration. Hence, it is an optimal scheme for a particular value of α = 2 according to theKung–Traub conjecture, completing the proof.


In this section, we illustrate the efficiency and convergence behavior of our iteration functionsfor particular values α = 0, α = 1, α = 1.9, and α = 2 in Expression (4), called OM1, OM2, OM3,and OM4, respectively. In this regards, we choose five real problems having multiple and simple zeros.The details are outlined in the examples (1)–(3).

For better comparison of our iterative methods, we consider several existing methods of order sixand the optimal order eight. Firstly, we compare our methods with the two-point family of sixth-ordermethods proposed by Geum et al. in [18], and out of them, we pick Case 4c, which is mentionedas follows:

yn = xn −mf (xn)

f ′(xn), m > 1,

xn+1 = yn −[

m + a1un

1 + b1un + b2un2 ×1

1 + c1sn

]f (yn)

f ′(yn),

(16)

where:

128


a1 =2m

(4m4 − 16m3 + 31m2 − 30m + 13

)(m− 1) (4m2 − 8m + 7)

, b1 =4(2m2 − 4m + 3

)(m− 1) (4m2 − 8m + 7)

,

b2 = −4m2 − 8m + 34m2 − 8m + 7

, c1 = 2(m− 1),

un =

(f (yn)

f (xn)

) 1m , sn =

(f ′(yn)

f ′(xn)

) 1m− 1 ,

called GM1.In addition, we also compare them with one more non-optimal family of sixth-order iteration

functions given by the same authors of [19], and out of them, we choose Case 5YD, which is given by:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

wn = xn −m[(un − 2) (2un − 1)(un − 1) (5un − 2)

]f (xn)

f ′(xn),

xn+1 = xn −m[

(un − 2) (2un − 1)(5un − 2) (un + vn − 1)

]f (xn)

f ′(xn),

(17)

where un =(

f (yn)f (xn)

) 1m and vn =

(f (wn)f (xn)

) 1m , and this method is denoted as GM2.

Moreover, we compare our methods with the optimal eighth-order iterative methods proposedby Zafar et al. [21]. We choose the following two schemes out of them:

yn = xn −mf (xn)

f ′(xn),

wn = yn −mun(6u3

n − u2n + 2un + 1

) f (xn)

f ′(xn),

xn+1 = wn −munvn(1 + 2un)(1 + vn)

(2wn + 1

A2P0

)f (xn)

f ′(xn)

(18)

and:

yn = xn −mf (xn)

f ′(xn),

wn = yn −mun

(1− 5u2

n + 8u3n

1− 2un

)f (xn)

f ′(xn),

xn+1 = wn −munvn(1 + 2un)(1 + vn)

(3wn + 1

A2P0(1 + wn)

)f (xn)

f ′(xn),

(19)

where un =(

f (yn)f (xn)

) 1m , vn =

(f (wn)f (yn)

) 1m , wn =

(f (wn)f (xn)

) 1m , and these iterative methods are denoted in

our tables as ZM1 and ZM2, respectively.Finally, we demonstrate their comparison with another optimal eighth-order iteration function

given by Behl et al. [22]. However, we consider the following the best schemes (which was claimedby them):

yn = xn −mf (xn)

f ′(xn),

zn = yn −mf (xn)

f ′(xn)hn(1 + 2hn),

xn+1 = zn + mf (xn)

f ′(xn)

tnhn

1− tn

[− 1− 2hn − h2

n + 4h3n − 2kn

] (20)

129


and:

yn = xn −mf (xn)

f ′(xn),

zn = yn −mf (xn)

f ′(xn)hn(1 + 2hn),

xn+1 = zn −mf (xn)

f ′(xn)

tnhn

1− tn

[1 + 9h2n + 2kn + hn(6 + 8kn)

1 + 4hn

],

(21)

with hn =(

f (yn)f (xn)

) 1m , kn =

(f (zn)f (xn)

) 1m tn =

(f (zn)f (yn)

) 1m , which are denoted BM1 and BM2, respectively.

In order to compare these schemes, we perform a numerical experience, and in Tables 1 and 2,we display the difference between two consecutive iterations |xn+1 − xn|, the residual error in thecorresponding function | f (xn)|, and the computational order of convergence (ρ) (we used the formulagiven by Cordero and Torregrosa [25]:

ρ ≈ ln(| xk+1 − xk | / | xk − xk−1 |)ln(| xk − xk−1 | / | xk−1 − xk−2 |) (22)

We make our calculations with several significant digits (a minimum of 3000 significant digits) tominimize the round-off error. Moreover, the computational order of convergence is provided up tofive significant digits. Finally, we display the initial guess and approximated zeros up to 25 significantdigits in the corresponding example where an exact solution is not available.

All computations have been performed using the programming package Mathematica 11 withmultiple precision arithmetic. Further, the meaning of a(±b) is shorthand for a × 10(±b) in thenumerical results.

Example 1. Population growth problem:The law of population growth is defined as follows:

dN(t)dt

= γN(t) + η,

where N(t) = the population at time t, η = the fixed/constant immigration rate, and γ = the fixed/constant birthrate of the population. We can easily obtain the following solution of the above differential equation:

N(t) = N0eγt +η

γ(eγt−1),

where N0 is the initial population.For a particular case study, the problem is given as follows: Suppose a certain population contains 1,000,000

individuals initially, that 300,000 individuals immigrate into the community in the first year, and that 1,365,000individuals are present at the end of one year. Find the birth rate (γ) of this population.

To determine the birth rate, we must solve the equation:

f1(x) = 1365− 1000ex − 300x

(ex − 1). (23)

wherein x = γ and our desired zero of the above function f1 is 0.05504622451335177827483421. The reasonfor considering the simple zero problem is to confirm that our methods also work for simple zeros. We choose thestarting point as x0 = 0.5.

130


Example 2. The van der Waals equation of state:(P +

a1n2

V2

)(V − na2) = nRT,

explains the behavior of a real gas by introducing in the ideal gas equations two parameters, a1 and a2, specificfor each gas. The determination of the volume V of the gas in terms of the remaining parameters requires thesolution of a nonlinear equation in V,

PV3 − (na2P + nRT)V2 + a1n2V − a1a2n2 = 0.

Given the constants a1 and a2 of a particular gas, one can find values for n, P, and T, such that thisequation has three simple roots. By using the particular values, we obtain the following nonlinear function:

f2(x) = x3 − 5.22x2 + 9.0825x− 5.2675. (24)

which has three zeros; out of them, one is the multiple zero α = 1.75 of multiplicity two, and the other is thesimple zero α = 1.72. Our desired root is α = 1.75, and we chose x0 = 1.8 as the initial guess.

Example 3. Eigenvalue problem:For this, we choose the following 8× 8 matrix:

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−12 −12 36 −12 0 0 12 8148 129 −397 147 −12 6 −109 −7472 62 −186 66 −8 4 −54 −36−32 −24 88 −36 0 0 24 1620 13 −45 19 8 6 −13 −10

120 98 −330 134 −8 24 −90 −60−132 −109 333 −115 12 −6 105 66

0 0 0 0 0 0 0 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The corresponding characteristic polynomial of this matrix is as follows:

f3(x) = (x− 4)3(x + 4)(x− 8)(x− 20)(x− 12)(x + 12).

The above function has one multiple zero at α = 4 of multiplicity three. In addition, we consider x0 = 2.7as the starting point.

Example 4. Let us consider the following polynomial equation:

f4(z) =((x− 1)3 − 1

)50. (25)

The desired zero of the above function f4 is α = 2 with multiplicity of order 50, and we choose initial guessx0 = 2.1 for this problem.

131


Ta

ble

1.

Com

pari

son

onth

eba

sis

ofth

edi

ffer

ence

betw

een

two

cons

ecut

ive

iter

atio

ns|x n

+1−

x n|fo

rth

efu

ncti

ons

f 1–

f 4.

fn

OM

1O

M2

OM

3O

M4

GM

1G

M2

ZM

1Z

M2

BM

1B

M2

f 1

12.

3(−

3)8.

4(−

4)9.

3(−

5)3.

5(−

5)*

3.6

(−5)

1.6

(−4)

2.3

(−4)

7.6

(−5)

3.7

(−5)

22.

0( −

16)

9.0

(−20

)8.

8(−

28)

2.0

(−37

)*

1.4

(−29

)4.

2(−

31)

8.9

(−30

)2.

6(−

34)

5.0

(−37

)3

9.7

( −95

)1.

3(−

115)

6.4

(−16

6)2.

5(−

295)

*5.

4(−

173)

1.0

(−24

3)5.

5(−

233)

5.4

(−27

0)5.

7(−

292)

ρ5.

9997

6.00

006.

0001

8.00

00*

6.00

008.

0000

8.00

008.

0000

8.00

00

f 2

11.

3(−

3)8.

2(−

4)4.

0(−

3)3.

5(−

4)9.

5(−

4)3.

9(−

4)3.

9(−

4)4.

1(−

3)2.

7(−

4)2.

6(−

4)2

2.5

(−10

)4.

2(−

12)

6.4

(−16

)8.

7(−

18)

2.7

(−11

)1.

0(−

14)

5.2

(−17

)9.

8(−

17)

1.1

(−18

)1.

4(−

19)

32.

0( −

50)

8.7

(−62

)6.

5(−

87)

1.5

(−12

6)2.

0(−

56)

3.9

(−78

)5.

9(−

120)

1.2

(−11

7)6.

3(−

134)

1.0

(−14

1)ρ

5.97

575.

9928

6.02

147.

9963

5.98

365.

9975

7.99

457.

9941

7.99

718.

0026

f 3

19.

1(−

5)3.

6(−

5)8.

0(−

6)6.

0(−

6)8.

5(−

5)4.

8(−

5)4.

9(−

6)5.

2(−

6)2.

0(−

6)1.

8(−

6)2

1.8

( −28

)1.

4(−

31)

9.8

(−38

)2.

0(−

47)

1.0

(−28

)5.

0(−

31)

6.0

(−48

)1.

0(−

47)

1.5

(−51

)2.

8(−

52)

31.

2( −

170)

4.4

(−19

0)3.

3(−

229)

2.5

(−37

9)3.

1(−

172)

5.8

(−18

7)2.

7(−

383)

2.3

(−38

1)1.

4(−

412)

1.3

(−41

8)ρ

6.00

006.

0000

6.00

008.

0000

6.00

006.

0000

8.00

008.

0000

8.00

008.

0000

f 4

12.

4(−

5)7.

1(−

6)4.

2(−

7)1.

4(−

7)1.

8(−

5)2.

0(−

7)4.

8(−

7)6.

5(−

7)1.

9(−

7)6.

3(−

8)2

1.5

( −26

)1.

7(−

30)

3.9

(−40

)6.

7(−

54)

1.1

(−26

)1.

8(−

41)

5.7

(−49

)8.

4(−

48)

8.0

(−53

)4.

2(−

57)

37.

5( −

154)

3.2

(−17

8)2.

6(−

438)

1.7

(−42

4)6.

6(−

154)

1.0

(−24

5)2.

2(−

384)

6.6

(−37

5)9.

6(−

416)

5.9

(−16

9)ρ

6.00

006.

0000

6.00

008.

0000

6.00

006.

0000

8.00

008.

0000

8.00

002.

2745

*m

eans

that

the

corr

espo

ndin

gm

etho

ddo

esno

twor

k.

Ta

ble

2.

Com

pari

son

onth

eba

sis

ofre

sidu

aler

rors|f(

x n)|

for

the

func

tion

sf 1

–f 4

.

fn

OM

1O

M2

OM

3O

M4

GM

1G

M2

ZM

1Z

M2

BM

1B

M2

f 1

12.

71.

01.

1(−

1)4.

2(−

2)*

4.4

(−2)

1.9

(−1)

2.7

(−1)

9.2

(−2)

4.4

(−2)

22.

4( −

13)

1.1

(−16

)1.

1(−

24)

2.4

(−34

)*

1.7

(−26

)5.

1(−

28)

1.1

(−26

)3.

2(−

31)

6.0

(−34

)3

1.2

( −91

)1.

6(−

112)

7.8

(−16

3)3.

0(−

292)

*5.

4(−

173)

1.2

(−24

0)6.

7(−

230)

6.5

(−26

7)7.

0(−

289)

f 2

15.

0(−

8)2.

1(−

8)4.

8(−

9)3.

6(−

9)2.

8(−

8)4.

6(−

9)4.

6(−

9)5.

1(−

9)2.

3(−

9)2.

0(−

9)2

1.8

( −21

)5.

3(−

25)

1.2

(−32

)2.

3(−

36)

2.2

(−23

)3.

2(−

30)

8.0

(−35

)2.

9(−

34)

3.4

(−38

)5.

9(−

40)

31.

2( −

101)

2.2

(−12

4)1.

3(−

174)

6.9

(−25

4)1.

2(−

113)

4.6

(−15

7)1.

1(−

240)

4.3

(−23

6)1.

2(−

268)

3.1

(−28

4)

f 3

14.

9(−

8)3.

1(−

9)3.

1(−

11)

1.4

(−11

)4.

1(−

8)7.

4(−

9)7.

8(−

12)

9.1

(−12

)5.

2(−

13)

3.6

(−13

)2

3.9

( −79

)1.

8(−

88)

6.1

(−10

7)4.

9(−

136)

7.1

(−80

)8.

0(−

87)

1.4

(−13

7)6.

9(−

137)

2.1

(−14

8)1.

5(−

150)

31.

0( −

505)

5.6

(−56

4)2.

4(−

681)

1.1

(−11

31)

1.9

(−51

0)1.

2(−

554)

1.3

(−11

43)

7.5

(−11

38)

1.9

(−12

31)

1.3

(−12

49)

f 4

11.

2(−

207)

2.7

(−23

4)1.

1(−

295)

3.3

(−31

9)3.

5(−

214)

1.0

(−31

1)6.

6(−

293)

2.3

(−28

6)1.

8(−

313)

6.2

(−33

7)2

1.9

( −12

68)

2.6

(−14

65)

3.8

(−19

47)

1.6

(−26

35)

1.9

(−12

74)

9.8

(−20

14)

3.4

(−23

89)

9.4

(−23

31)

9.8

(−25

82)

1.1

(−27

95)

34.

2( −

7633

)2.

3(−

8851

)7.

5(−

1185

6)6.

1(−

2116

6)6.

0(−

7636

)7.

3(−

1222

6)1.

6(−

1915

9)7.

1(−

1868

6)8.

8(−

2072

8)3.

4(−

8388

)

132


4. Conclusions

We presented an eighth-order modification of the Chebyshev–Halley-type iteration scheme havingoptimal convergence to obtain the multiple solutions of the scalar equation. The proposed scheme isoptimal in the sense of the classical Kung–Traub conjecture. Thus, the efficiency index of the presentmethods is E = 4

√8 ≈ 1.682, which is better than the classical Newton’s method E = 2

√2 ≈ 1.414.

Finally, the numerical experience corroborates the theoretical results about the convergence order,and moreover, it can be concluded that our proposed methods are highly efficient and competitive.

Author Contributions: First two authors have contribute to the theoretical results and the other two authors havecarried out the numerical experience.

Funding: This research was partially supported by Ministerio de Economía y Competitividad under GrantMTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089.


References

1. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.2. Gutiérrez, J.M.; Hernández, M.A. A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust.

Math. Soc. 1997, 55, 113–130. [CrossRef]3. Kanwar, V.; Singh, S.; Bakshi, S. Simple geometric constructions of quadratically and cubically convergent

iterative functions to solve nonlinear equations. Numer. Algorithms 2008, 47, 95–107. [CrossRef]4. Potra, F.A.; Pták, V. Nondiscrete Induction and Iterative Processes; Research Notes in Mathematics; Pitman:

Boston, MA, USA, 1984; Volume 103.5. Sharma, J.R. A composite third order Newton-Steffensen method for solving nonlinear equations.

Appl. Math. Comput. 2005, 169, 242–246.6. Argyros, I.K.; Ezquerro, J.A.; Gutiérrez, J.M.; Hernández, M.A.; Hilout, S. On the semilocal convergence of

efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 2011, 235, 3195–3206. [CrossRef]7. Xiaojian, Z. Modified Chebyshev–Halley methods free from second derivative. Appl. Math. Comput. 2008,

203, 824–827. [CrossRef]8. Ostrowski, A.M. Solutions of Equations and System of Equations; Academic Press: New York, NY, USA, 1960.9. Amat, S.; Hernández, M.A.; Romero, N. A modified Chebyshev’s iterative method with at least sixth order

of convergence. Appl. Math. Comput. 2008, 206, 164–174. [CrossRef]10. Kou, J.; Li, Y. Modified Chebyshev–Halley method with sixth-order convergence. Appl. Math. Comput. 2007,

188, 681–685. [CrossRef]11. Li, D.; Liu, P.; Kou, J. An improvement of Chebyshev–Halley methods free from second derivative.

Appl. Math. Comput. 2014, 235, 221–225. [CrossRef]12. Sharma, J.R. Improved Chebyshev–Halley methods with sixth and eighth order convergence.

Appl. Math. Comput. 2015, 256, 119–124. [CrossRef]13. Neta, B. Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 2010,

87, 1023–1031. [CrossRef]14. Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the multiple roots of nonlinear

equations. J. Comput. Math. Appl. 2011, 235, 4199–4206. [CrossRef]15. Hueso, J.L.; Martínez, E.; Teruel, C. Determination of multiple roots of nonlinear equations and applications.

J. Math. Chem. 2015, 53, 880-892. [CrossRef]16. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. On developing fourth-order optimal families of methods

for multiple roots and their dynamics. Appl. Math. Comput. 2015, 265, 520-532. [CrossRef]17. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R.; Kanwar, V. An optimal fourth-order family of methods for

multiple roots and its dynamics. Numer. Algorithms 2016, 71, 775–796. [CrossRef]18. Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modified

double-Newton type and their dynamics. Appl. Math. Comput. 2015, 270, 387–400. [CrossRef]19. Geum, Y.H.; Kim, Y.I.; Neta, B. A sixth-order family of three-point modified Newton-like multiple-root

finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 2016, 283, 120–140.[CrossRef]

133


20. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. An eighth-order family of optimal multiple root findersand its dynamics. Numer. Algorithms 2018, 77, 1249–1272. [CrossRef]

21. Zafar, F.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots ofnonlinear equations using free parameters. J. Math. Chem. 2018, 56, 1884–1901. [CrossRef]

22. Behl, R.; Alshomrani, A.S.; Motsa, S.S. An optimal scheme for multiple roots of nonlinear equations witheighth-order convergence. J. Math. Chem. 2018, 56, 2069–2084. [CrossRef]

23. Behl, R.; Zafar, F.; Alshomrani, A.S.; Junjuaz, M.; Yasmin, N. An optimal eighth-order scheme for multiplezeros of univariate functions. Int. J. Comput. Methods 2018, 1843002. [CrossRef]

24. McNamee, J.M. A comparison of methods for accelerating convergence of Newton’s method for multiplepolynomial roots. ACM Signum Newsl. 1998, 33, 17–22. [CrossRef]

25. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas.Appl. Math. Comput. 2007, 190, 686–698. [CrossRef]


134

mathematics

Article

An Optimal Eighth-Order Family of IterativeMethods for Multiple Roots

Saima Akram * , Fiza Zafar and Nusrat Yasmin

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University,Multan 60800, Pakistan* Correspondence: [email protected]

Received: 5 June 2019; Accepted: 25 July 2019; Published: 27 July 2019

Abstract: In this paper, we introduce a new family of efficient and optimal iterative methods forfinding multiple roots of nonlinear equations with known multiplicity (m ≥ 1). We use the weightfunction approach involving one and two parameters to develop the new family. A comprehensiveconvergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggestedscheme. Finally, numerical and dynamical tests are presented, which validates the theoretical resultsformulated in this paper and illustrates that the suggested family is efficient among the domain ofmultiple root finding methods.

Keywords: nonlinear equations; optimal iterative methods; multiple roots; efficiency index

1. Introduction

The problem of solving nonlinear equation is recognized to be very old in history as manypractical problems which arise are nonlinear in nature . Various one-point and multi-point methodsare presented to solve nonlinear equations or systems of nonlinear equations [1–3]. The above-citedmethods are designed for the simple root of nonlinear equations but the behavior of these methodsis not similar when dealing with multiple roots of nonlinear equations. The well known Newton’smethod with quadratic convergence for simple roots of nonlinear equations decays to first order whendealing with multiple roots of nonlinear equations. These problems lead to minor troubles such asgreater computational cost and severe troubles such as no convergence at all. The prior knowledge ofmultiplicity of roots make it simpler to manage these troubles. The strange behavior of the iterativemethods while dealing with multiple roots has been well known since 19th century in the least whenSchröder [4] developed a modification of classical Newton’s method to conserve its second order ofconvergence for multiple roots. The nonlinear equations with multiple roots commonly arise fromdifferent topics such as complex variables, fractional diffusion or image processing, applications toeconomics and statistics (Levy distributions), etc. By knowing the practical nature of multiple rootfinders, various one-point and multi-point root solvers have been developed in recent past [5–18]but most of them are not optimal as defined by Kung and Traub [19], who stated that an optimalwithout memory method can achieve its convergence order at the most 2n requiring n + 1 evaluationsof functions or derivatives. As stated by Ostrowski [1], if an iterative method possess order ofconvergence as O and total number of functional evaluations is n per iterative step, then the indexdefined by E = O1/n is recognized as efficiency index of an iterative method.

Sharma and Sharma [17] proposed the following optimal fourth-order multiple root finder withknown multiplicity m as follows:

yn = xn − 2mm+2 · f (xn)

f ′(xn), m > 1

xn+1 = xn − m8 Φ(xn)

f (xn)f ′(xn)

,



where Φ(xn) ={(m3 − 4m + 8)− (m + 2)2( m

m+2 )m f ′(xn)

f ′(yn)× 2(m− 1)(m + 2)( m

m+2 )m f ′(xn)

f ′(yn)

}.

A two-step sixth-order non-optimal family for multiple roots presented by Geum et al. [9] isgiven by:

yn = xn −m · f (xn)

f ′(xn), m > 1,

xn+1 = yn −Q(rn, sn) · f (yn)

f ′(yn), (1)

where, rn = m

√f (yn)f (xn)

, sn = m−1

√f ′(yn)f ′(xn)

and Q : C2 → C is holomorphic in a neighborhood of (0, 0).

The following is a special case of their family:


f ′(xn), n ≥ 0, m > 1,

xn+1 = yn −m[1 + 2(m− 1)(rn − sn)− 4rnsn + s2

n

]· f (yn)

f ′(yn). (2)

Another non-optimal family of three-point sixth-order methods for multiple roots byGeum et al. [10] is given as follows:


f ′(xn), m ≥ 1,

wn = yn −m · G(pn) · f (xn)

f ′(xn), (3)

xn+1 = wn −m · K(pn, vn, ) · f (xn)

f ′(xn),

where pn = m

√f (yn)f (xn)

and vn = m

√f (wn)f (xn)

. The weight functions G : C→ C is analytic in a neighborhood

of 0 and K : C2 → C is holomorphic in a neighborhood of (0, 0). The following is a special case of thefamily in Equation (3):


f ′(xn), m ≥ 1,

wn = yn −m ·[1 + pn + 2p2

n

]· f (xn)

f ′(xn), (4)

xn+1 = wn −m ·[1 + pn + 2p2

n + (1 + 2pn)vn

]· f (xn)

f ′(xn).

The families in Equations (1) and (3) require four evaluations of function to produce convergenceof order six having efficiency index 6

14 = 1.5650 and therefore are not optimal in the sense of the

Kung–Traub conjecture [19].Recently, Behl et al. [20] presented a multiple root finding family of iterative methods possessing

convergence order eight given as:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

zn = yn − unQ(hn)f (xn)

f ′(xn), (5)

xn+1 = zn − untnG(hn, tn)f (xn)

f ′(xn),

136


where the functions Q : C→ C and G : C2 → C are restricted to be analytic functions in the regions

nearby (0) and (0, 0), respectively, with un =(

f (yn)f (xn)

) 1m , hn = un

a1+a2unand tn =

(f (zn)f (yn)

) 1m , being a1

and a2 complex non-zero free parameters.We take Case (27) for (a1 = 1, a2 = 1, G02 = 0) from the family of Behl et al. [20] and represent it

by BM given by:

yn = xn −mf (xn)

f ′(xn),

zn = yn −(

m + 2hnm +12

h2n(4m + 2m)

)f (xn)

f ′(xn)un (6)

xn+1 = zn −(

m + mtn + 3mh2n + mhn(2 + 4tn + hn)

) f (xn)

f ′(xn)untn.

Most recently, another optimal eighth-order scheme presented by Zafar et al. [21] is given as:

yn = xn −mf (xn)

f ′(xn), m ≥ 1,

zn = yn −mun H(un)f (xn)

f ′(xn), (7)

xn+1 = zn − untn(B1 + B2un)P(tn)G(wn)f (xn)

f ′(xn),

where B1,B2 ∈ R are suppose to be free parameters and weight functions H : C→ C, P : C→ C and

G : C → C are restricted to be analytic in the regions nearby 0 with un =(

f (yn)f (xn)

) 1m , tn =

(f (zn)f (yn)

) 1m

and wn =(

f (zn)f (xn)

) 1m .

From the eighth-order family of Zafar et al. [21], we consider the following special case denotedby ZM:

yn = xn −mf (xn)

f ′(xn),

zn = yn −mun

(6u3

n − u2n + 2un + 1

) f (xn)

f ′(xn),

xn+1 = zn −muntn (1 + 2un) (1 + tn)(1 + 2wn)f (xn)

f ′(xn). (8)

The class of iterative methods referred as optimal is significant as compared to non-optimalmethods due to their speed of convergence and efficiency index. Therefore, there was a need todevelop optimal eighth-order schemes for finding multiple zeros (m > 1) and simple zeros (m = 1)due to their competitive efficiencies and order of convergence [1]; in addition, fewer iterations areneeded to get desired accuracy as compared to iterative methods having order four and six given bySharma and Geum [9,10,17], respectively. In this paper, our main concern is to find the optimal iterativemethods for multiple root μ with known multiplicity m ∈ N of an adequately differentiable nonlinearfunction f : I ⊆ R→ R, where I represents an open interval. We develop an optimal eighth-order zerofinder for multiple roots with known multiplicity m ≥ 1. The beauty of the method lies in the fact thatdeveloped scheme is simple to implement with minimum possible number of functional evaluations.Four evaluations of the function are needed to obtain a family of convergence order eighth havingefficiency index 8

14 = 1.6817.

The rest of the paper is organized as follows. In Section 2, we present the newly developedoptimal iterative family of order eight for multiple roots of nonlinear equations. The discussion ofanalysis of convergence is also given in this section. In Section 3, some special cases of newly developed

137


eighth-order schemes are presented. In Section 4, numerical results and comparison of the presentedschemes with existing schemes of its domain is discussed. Concluding remarks are given in Section 5.

2. Development of the Scheme and Convergence Analysis

In this section, we suggest a new family of eighth-order method with known multiplicity m ≥ 1of the required multiple root as follows:


f ′(xn), n ≥ 0,

zn = yn −m · t · H(t) · f (xn)

f ′(xn), (9)

xn+1 = zn −m · t · L(s, u) · f (xn)

f ′(xn),

where t = m

√f (yn)

f (xn), s = m

√f (zn)

f (yn), u = m

√f (zn)

f (xn).

where the function H : C → C is restricted to be analytic function in the regions nearby 0 andweight function L : C2 → C is holomorphic in the regions nearby (0, 0) and t, s and u are one-to-mmultiple-valued functions.

In the next theorem, it is demonstrated that the proposed scheme in Equation (9) achieves theoptimal eighth order of convergence without increasing the number of functional evaluations.

Theorem 1. Suppose x = μ (say) is a multiple root having multiplicity m ≥ 1 of an analytic functionf : C→ C in the region enclosing a multiple zero μ of f (x). Which implies that the family of iterative methodsdefined by Equation (9) has convergence of order eighth when the following conditions are fulfilled:

H0 = 1, H1 = 2, H2 = −2, H3 = 36, L00 = 0, L10 = 1, L01 = 2, L11 = 4, L20 = 2. (10)

Then, the proposed scheme in Equation (9) satisfies the following error equation:

en+1 =1

24m7 {c1(c21(11 + m)− 2mc2)((677 + 108m + 7m2)c4

1

−24m(9 + m)c21c2 + 12m2c2

2 + 12m2c1c3)e8n}+ O(e9

n), (11)

where en = xn − μ and ck =m!

(m+k)!f (m+k)(μ)

f (m)(μ), k = 1, 2, 3, · · ·.

Proof. Suppose x = μ is a multiple root of f (x). We expand f (xn) and f ′(xn) by Taylor’s seriesexpansion about x = μ using Mathematica (Computer based algebra software), to get

f (xn) =f (m)(μ)

m!em

n

(1 + c1en + c2e2

n + c3e3n + c4e4

n + c5e5n + c6e6

n + c7e7n + c8e8

n + O(e9n))

, (12)

and

f ′(xn) = f (m)(μ)m! em−1

n m + c1(m + 1)en + c2(m + 2)e2n + c3(m + 3)e3

n + c4(m + 4)e4n

+c5(m + 5)e5n + c6(m + 6)e6

n + c7(m + 7)e7n + c8(m + 8)e8

n + O(e9n),

respectively. By utilizing the above Equations (11) and (12) in the first substep of Equation (9), we obtain

yn − μ =c1e2

nm

+(2c2m− c2

1(m + 1))e3n

m2 + ∑4

limk=0

Gkek+4n + O(e9

n), (13)

138


where Gk = Gk(m, c1, c2, . . . , c8) are expressed in terms of m, c1, c2, c3, . . . , c8 and the two coefficientsG0and G1 can be explicitly written as G0 = 1

m3 {3c3m2 + c31(m + 1)2 − c1c2m(3m + 4)} and G1 =

− 1m4 {c4

1(m + 1)3 − 2c2c21m(2m2 + 5m + 3) + 2c3c1m2(2m + 3) + 2m2(c2

2(m + 2)− 2c4m)}. By Taylor’sexpansion, we get

f (yn) = f (m)(μ)e2mn

[( c1

m )m

m!+

(2mc2 − (m + 1)c21)(

c1m )men

c1m!+

6

∑k=0

Gkek+2n + O(e9

n)

]. (14)

By using Equations (12) and (14), we get

u =c1en

m+

(2mc2 − (m + 2)c21)e

2n

m2 + ψ1e3n + ψ2e4

n + ψ3e5n + O(e6

n), (15)

where ψ1 = 12m3 [c3

1(2m2 + 7m + 7) + 6c3m2 − 2c1c2m(3m + 7)], ψ2 = − 16m4 [c4

1(6m3 + 29m2 +

51m + 34) − 6c2c21m(4m2 + 16m + 17) + 12c1c3m2(2m + 5) + 12m2(c2

2(m + 3) − 2c4m)], ψ3 =1

24m5 [−24m3(c2c3(5m + 17)− 5c5m) + 12c3c21m2(10m2 + 43m + 49) + 12c1m2{c2

2(10m2 + 47m + 53)−2c4m(5m + 13)} − 4c2c3

1m(30m3 + 163m2 + 306m + 209) + c51(24m4 + 146m3 + 355m2 + 418m + 209)].

Taylor series of H(t) about 0 is given by:

H(t) = H0 + H1t +H2

2!t2 +

H3

3!t3 + O(e4

n) (16)

where Hj = Hj(0) for 0 ≤ j ≤ 3. Inserting Equations (13)–(16) in the second substep of the scheme inEquation (9), we get

zn = μ +−(1 + H0)c1e2

nm

− (1 + H1 + m− H0(3 + m)c21) + 2(−1 + H0)mc2)e3

nm2

+1

2m3

[(2 + 10H1 − H2 + 4m + 4H1m + 2m2 − H0(13 + 11m + 2m2))c3

1

+2m(−4− 4H1 − 3m + H0(11 + 3m)c1c2 − 6(−1 + H0)m2c3)e4n

]+ z5e5

n

+z6e6n + z7e7

n + O(e8n).

By selecting H0 = 1 and H1 = 2, we obtain

zn = μ +(c3

1(9− H2 + m)− 2mc1c2)

2m3 e4n + z5e5

n + z6e6n + z7e7

n + O(e8n), (17)

where z5 = − 16m4 {c4

1(125 + H3 + 84m + 7m2 − 3H2(7 + 3m) + 6m(−3H2 + 4(7 + m))c21c2 + 12c2

2m2 +

12c2c1m) , z6 = 124m5 {1507 + 1850m + 677m2 + 46m3 + 4H3(9 + 4m)− 6H2(59 + 53m + 12m2))c5

1 −4m(925 + 8H3 + 594m + 53m2 − 3H2(53 + 21m)c3

1c2 +12m2(83 − 9H2 + 13m)c21c3 − 168m3c2c3 +

12m2c1(115− 12H2 + 17m)c22 − 6mc4) and z7 = −{12c2

1c3m2(36β + 13m + 11) + (37− 168c2c3m3 +

4c31c2m(96β2 + 252β + 53m2 + 18(14β + 5)m) + 12c1m2(c2

2(48β + 17m + 19)− 6c4m)}.Again, we use the Taylor’s expansion for Equation (17) to get:

f (zn) = f (m)(μ)e4mn

2−m(

c31(9−H2+m)−2mc1c2

m3

)m

m! −

(2−m

(c31(9−H2+m)−2mc1c2

m3

)m−1

ρ0

)3(m3m!) en

+∑ lim7j=0 Hje

j+1n + O(e9

n),

139


where ρ0 = c41(125 + H3 + 84m + 7m2 − 3H2(7 + 3m))c4

1 − 6m(−3H2 + 4(7 + m))c21c2 + 12m2c2

2 +

12c3c1m2). With the help of Equations (12) and (18), we have

s =c2

1(9− H2 + m)− 2mc2

2m2 e2n + ρ1e3

n + ρ2e4n + ρ3e5

n + O(e6n), (18)

where

ρ1 = − 16m3 {c3

1(98 + H3 + 4m2 + 54m− 6H2(3 + m)− 12m(9− H2 + m)c1c2 + 12m2c3},

ρ2 =1

24m4 899 + 1002m + 313m2 + 18m3 + 4H3(8 + 3m)− 6H2(43 + 33m + 6m2))c41−

12m(167 + 2H3 + 87m + 6m2 − H2(33 + 10m)c21c2 + 24m2(26− 3H2 + 3m)c1c3+

12m2(c22(35− 4H2 + 3m)− 6mc4)

and ρ3 = − 160m5 [−4257− 7270m − 4455m2 − 101m3 − 48m4 − 10H3(37 + 30m + 6m2) + 30H2(60 +

75m + 31m2 + 4m3)c51 + 10m(1454 + 60H3 + 1548m + 21H3m + 454m2 + 24m3 − 18H2(25 + 18m +

3m2)c31c2 − 30m2(234 + 3H3 + 118m + 8m2 − 2H2(24 + 7m)c2

1c3 − 60m2c1(141 + 2H3 + 67m +

4m2 − 2H2(15 + 4m)c22 + 2(−17 + 2H2 − 2m)mc4) − 120m3(−25 + 3H2 − 2m)c2c3 + 2mc5} +

( 1720m6 )((102047+ 180H2

2 + 204435m + 187055m2 + 81525m3 + 14738m4 + 600m5 + 40H3(389+ 498m +

214m2 + 30m3)− 45H2(1223+ 2030m+ 1353m2 + 394m3 + 40m4))− 30m(13629+ 22190m+ 12915m2 +

2746m3 + 120m4 + 16H3(83 + 64m + 12m2)− 6H2(1015 + 1209m + 470m2 + 56m3)) + 120m2(2063 +

2088m + 589m2 + 30m3 + H3(88+ 30m)− 18H2 + (36+ 25m + 4m2)) + 80m2(2323+ 2348m + 635m2 +

30m3 + 4H3(28+ 9m)− 3H2(259+ 173m + 26m2))− 2m(303+ 4H3 + 149m + 10m2− 9H2(7+ 2m))−720m3((393+ 6H3 + 178m+ 10m2−H2(87+ 22m))] + (−42+ 5H2− 5m)mc5) + 20m3((−473− 8H3−195m− 10m2 + 12H2(9 + 2m))c2c3 + 6m(65− 8H2 + 5m)c2 + 3m(71− 9H2 + 5m)c10mc6.

Since it is obvious from Equation (15) that u possess order en, the expansion of weight functionL f (s, u) by Taylor’s series is possible in the regions nearby origin given as follows:

L(s, u) = L00 + sL10 + uL01 + suL11 +s2

2!L20 (19)

where Li,j =1

i!j!∂i+j

∂sj∂uj L(s, u)∣∣∣(0,0)

. By using Equations (12)–(19) in the proposed scheme in Equation (9),

we haveen+1 = M2e2

n + M3e3n + M4e4

n + M5e5n + M6e6

n + M7e7n + O(e8

n), (20)

where the coefficients Mi(2 ≤ i ≤ 7) depend generally on m and the parameters Li,j. To obtain at leastfifth-order convergence, we have to choose L00 = 0, L10 = 1 and get

en+1 =((−2 + L01)c2

1((−9 + H2 −m)c21 + 2mc2)

2m4 e5n + M6e6

n + M7e7n + O(e8

n).

where the coefficients Mi(6 ≤ i ≤ 7) depend generally on m and the parameters Li,j. To obtaineighth-order convergence, we are restricted to choosing the values of parameters given by:

H2 = −2, H3 = 36, L00 = 0, L10 = 1, L01 = 2, L20 = 2, L11 = 4. (21)

This leads us to the following error equation:

en+1 =1

24m7 [c1(c21(11 + m)− 2mc2)((677 + 108m + 7m2)c4

1 − 24m(9 + m)c21c2

+12m2c22 + 12m2c1c3)]e8

n + O(e9n). (22)

140


The above error equation (Equation (22)) confirms that the presented scheme in Equation (9)achieves optimal order of convergence eight by utilizing only four functional evaluations (usingf (xn), f ′(xn), f (yn) and f (zn)) per iteration.

3. Special Cases of Weight Functions

From Theorem 1, several choices of weight functions can be obtained. We have consideredthe following:

Case 1: The polynomial form of the weight function satisfying the conditions in Equation (10) can berepresented as:

H(t) = 1 + 2t− t2 + 6t3

L(s, u) = s + 2u + 4su + s2 (23)

The particular iterative method related to Equation (23) is given by:SM-1:


f ′(xn), n ≥ 0,

zn = yn −m · t · (1 + 2t− t2 + 6t3)f (xn)

f ′(xn),

xn+1 = zn −m · t · (s + s2 + 2u + 4su) · f (xn)

f ′(xn)

where t = m

√f (yn)

f (xn), s = m

√f (zn)

f (yn), u = m

√f (zn)

f (xn)(24)

Case 2: The second suggested form of the weight functions in which H(t) is constructed usingrational weight function satisfying the conditions in Equation (10) is given by:

H(t) =1 + 8t + 11t2

1 + 6tL(s, u) = s + 2u + 4su + s2 (25)

The corresponding iterative method in Equation (25) can be presented as:SM-2:


f ′(xn), n ≥ 0,

zn = yn −m · t · (1 + 8t + 11t2

1 + 6t)

f (xn)

f ′(xn),

xn+1 = zn −m · t · (s + s2 + 2u + 4su) · f (xn)

f ′(xn)

where t = m

√f (yn)

f (xn), s = m

√f (zn)

f (yn), u = m

√f (zn)

f (xn)(26)

Case 3: The third suggested form of the weight function in which H(t) is constructed usingtrigonometric weight satisfying the conditions in Equation (10) is given by:

H(t) =5 + 18t

5 + 8t− 11t2

L(s, u) = s + 2u + 4su + s2 (27)

141


The corresponding iterative method obtained using Equation (27) is given by:SM-3:


f ′(xn), n ≥ 0,

zn = yn −m · t · ( 5 + 18t5 + 8t− 11t2 )

f (xn)

f ′(xn),

xn+1 = zn −m · t · (s + s2 + 2u + 4su) · f (xn)

f ′(xn)

where t = m

√f (yn)

f (xn), s = m

√f (zn)

f (yn), u = m

√f (zn)

f (xn). (28)

4. Numerical Tests

In this section, we show the performance of the presented iterative family in Equation (9) bycarrying out some numerical tests and comparing the results with existing method for multiple roots.All numerical computations were performed in Maple 16 programming package using 1000 significantdigits of precision. When μ was not exact, we preferred to take the accurate value which has largernumber of significant digits rather than the assigned precision. The test functions along with theirroots μ and multiplicity m are listed in Table 1 [22]. The proposed methods SM-1 (Equation (24)), SM-2(Equation (26)) and SM-3 (Equation (28)) are compared with the methods of Geum et al. given inEquations (2) and (4) denoted by GKM-1 and GKM-2 and with method of Bhel given in Equation (6)denoted by BM and Zafar et al. method given in Equation (8) denoted by ZM. In Tables 1–8, the errorin first three iterations with reference to the sought zeros (|xn − μ|) is considered for different methods.The notation E(−i) can be considered as E× 10−i. The test function along with their initial estimatesx0 and computational order of convergence (COC) is also included in these tables, which is computedby the following expression [23]:

COC ≈ log |(xk+1 − μ)/(xk − μ)|log |(xk − μ)/(xk−1 − μ)| .

Table 1. Test functions.

Test Functions Exact Root μ Multiplicity m

f1(x) = (cos(πx2 ) + x2 − π)5 2.034724896... 5

f2(x) = (ex + x− 20)2 2.842438953... 2f3(x) = (ln x +

√(x4 + 1)− 2)9 1.222813963... 9

f4(x) = (cosx− x)3 0.7390851332... 3f5(x) = ((x− 1)3 − 1)50 2.0 50f6(x) = (x3 + 4x2 − 10)6 1.365230013... 6

f7(x) = (8xe−x2 − 2x− 3)8 −1.7903531791... 8

Table 2. Comparison of different methods for multiple roots.

f1(x), x0 = 2.5

GKM-1 GKM-2 SM-1 SM-2 SM-3 ZM BM

|x1 − μ| 6.83(−4) 1.11(−3) 2.15(−4) 1.87(−4) 2.03(−4) 1.52(−4) 1.84(−4)|x2 − μ| 3.42(−14) 2.53(−18) 2.37(−29) 3.53(−30) 1.25(−29) 9.69(−31) 2.89(−30)|x3 − μ| 2.13(−55) 3.58(−106) 5.28(−229) 5.71(−236) 2.53(−231) 2.56(−240) 1.05(−236)

COC 4.00 6.00 8.00 8.00 8.00 8.00 8.00

142



f2(x), x0 = 3.0


|x1 − μ| 1.18(−7) 5.27(−6) 2.33(−7) 1.21(−7) 1.90(−7) 1.40(−7) 1.16(−7)|x2 − μ| 2.62(−37) 1.15(−32) 1.30(−53) 2.21(−56) 1.99(−54) 1.30(−55) 1.57(−56)|x3 − μ| 3.07(−221) 1.25(−192) 1.19(−423) 2.67(−446) 2.87(−430) 7.37(−440) 1.73(−447)

COC 6.00 6.00 8.00 8.00 8.00 8.00 8.00


f3(x), x0 = 3.0


|x1 − μ| 5.50(−1) 4.29(−2) 1.81(−2) 1.75(−2) 1.79(−2) D * D|x2 − μ| 3.99(−7) 8.77(−10) 2.82(−15) 9.58(−16) 2.04(−15) D D|x3 − μ| 1.13(−27) 7.51(−56) 2.06(−117) 8.21(−122) 6.49(−119) D D

COC 4.00 6.00 8.00 8.00 8.00 D D

* D stands for divergence.


f4(x), x0 = 1.0

GKM-1 GKM-2 SM-1 SM-2 SM-3 ZM BM|x1 − μ| 2.77(−4) 2.55(−5) 6.78(−8) 5.45(−8) 6.29(−8) 4.90(−8) 5.15(−8)|x2 − μ| 3.28(−14) 6.83(−36) 7.95(−60) 8.55(−61) 3.83(−60) 4.06(−61) 4.91(−61)|x3 − μ| 5.86(−49) 2.51(−213) 2.82(−475) 3.11(−483) 7.18(−478) 8.99(−486) 3.36(−485)

COC 3.50 6.00 8.00 8.00 8.00 7.99 7.99


f5(x), x0 = 2.1


COC 3.99 6.00 8.00 8.00 8.00 7.99 7.99


f6(x), x0 = 3.0


COC 3.97 5.96 8.00 7.98 7.97 7.97 7.97


f7(x), x0 = −1.2


|x1 − μ| 2.65(−3) 2.15(−3) 4.38(−4) 4.24(−4) 4.32(−4) 3.41(−4) 4.26(−4)|x2 − μ| 7.24(−12) 9.63(−17) 4.44(−27) 1.11(−27) 3.11(−27) 3.58(−28) 1.14(−27)|x3 − μ| 4.05(−46) 7.81(−97) 4.97(−211) 2.55(−216) 2.28(−212) 5.27(−220) 3.06(−216)

COC 4.00 6.00 8.00 8.00 8.00 7.99 7.99

143


It is observed that the performance of the new method SM-2 is the same as BM for the function f1

and better than ZM for the function f2. The newly developed schemes SM-1, SM-2 and SM-3 are notonly convergent but also their speed of convergence is better than GKM-1 and GKM-2 while ZM andBM show divergence for the function f3. For functions f4, f5, f6 and f7, the newly developed schemesSM-1, SM-2 and SM-3 are comparable with ZM and BM. Hence, we conclude that the proposed familyis comparable and robust among existing methods for multiple roots.

5. Dynamical Analysis

For the sake of stability comparison, we plot the dynamical planes corresponding to each scheme(SM-1, SM-2, SM-3, BM and ZM) for the nonlinear functions f1, f2, f3, f4, f5, f6, f7 by using the proceduredescribed in [24]. We draw a mesh of 400 × 400 points such that each point of the mesh is aninitial-approximation of the required root of corresponding nonlinear function. The point is orange ifthe sequence of iteration method converges to the multiple root (with tolerance 10−3) in fewer than 80iterations and the point is black if the sequence does not converges to the multiple root. The multiplezero is represented by a white star in the figures. Figures 1–14 show that the basin of attraction drawnin orange is of the multiple zero only (i.e., a set of initial guesses converging to the multiple roots fillsall the plotted regions of the complex plane). In general, convergence to other zeros or divergencecan appear (referred to as strange stationary points). SM-1 has wider regions of convergence for f1 ascompared to ZM and BM in Figures 1 and 2; SM-1 and SM-3 have wider regions of convergence forf2 as compared to ZM and BM in Figures 3 and 4. The convergence region of SM-2 for functions f3,f4 and f6 is comparable with ZM and BM, as shown in Figures 5–8, 11 and 12. For function f5 inFigures 9 and 10, the convergence region of SM-3 is better than ZM and BM. For function f7, SM-1 andSM-3 have better convergence regions than ZM and BM, as shown in Figures 13 and 14. Figures 1–14show that the region in orange is comparable or bigger for the presented methods SM-1, SM-2 andSM-3 than the regions obtained by schemes BM and ZM, which confirms the fast convergence andstability of the proposed schemes.

Figure 1. Basins of attraction of SM1 (Left), SM2 (Middle) and SM3 (Right) for f1(x).

Figure 2. Basins of attraction of BM (Left) and ZM (Right) for f1(x).

144







145







146




6. Conclusions

In this paper, we present a new family of optimal eighth-order schemes to find multiple roots ofnonlinear equations. An extensive convergence analysis is done, which verifies that the new familyis optimal eighth-order convergent. The presented family required four functional evaluations toget optimal eighth-order convergence, having efficiency index 8

14 = 1.6817, which is higher than the

efficiency index of the methods for multiple roots and of the families of Geum et al. [9,10]. Finally,numerical and dynamical tests confirmed the theoretical results and showed that the three membersSM-1, SM-2 and SM-3 of the new family are better than existing methods for multiple roots. Hence,the proposed family is efficient among the domain of multiple root finding methods.

Author Contributions: Methodology, S.A.; writing—original draft preparation, S.A.; investigation, S.A.;writing—review and editing, F.Z. and N.Y.; and supervision, F.Z. and N.Y.



References

1. Ostrowski, A.M. Solution of Equations and Systems of Equations; Academic Press: New York, NY, USA, 1960.2. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Dzunic, J. Multipoint Methods for Solving Nonlinear Equations;

Academic Press: New York, NY, USA, 2013.3. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.4. Schröder, E. Über unendlich viele algorithmen zur auflösung der gleichungen. Math. Annal. 1870, 2, 317–365.

[CrossRef]5. Singh, A.; Jaiswal, P. An efficient family of optimal fourth-order iterative methods for finding multiple roots

of nonlinear equations. Proc. Natl. Acad. Sci. India Sec. A 2015, 85, 439–450. [CrossRef]6. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R.; Kanwar, V. An optimal fourth-order family of methods for

multiple roots and its dynamics. Numer. Algorithms 2016, 71, 775–796. [CrossRef]7. Biazar, J.; Ghanbari, B. A new third-order family of nonlinear solvers for multiple roots. Comput. Math. Appl.

2010, 59, 3315–3319. [CrossRef]8. Chun, C.; Bae, H.J.; Neta, B. New families of nonlinear third-order solvers for finding multiple roots.

Comput. Math. Appl. 2009, 57, 1574–1582. [CrossRef]9. Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modified

double-Newton type and their dynamics. Appl. Math. Comput. 2015, 270, 387–400. [CrossRef]

147


10. Geum, Y.H.; Kim, Y.I.; Neta, B. A sixth-order family of three-point modified Newton-like multiple-rootfinders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 2016, 283, 120–140.[CrossRef]

11. Hueso, J.L.; Martinez, E.; Treuel, C. Determination of multiple roots of nonlinear equation and applications.J. Math. Chem. 2015, 53, 880–892. [CrossRef]

12. Lee, S.; Choe, H. On fourth-order iterative methods for multiple roots of nonlinear equation with highefficiency. J. Comput. Anal. Appl. 2015, 18, 109–120.

13. Lin, R.I.; Ren, H.M.; Šmarda, Z.; Wu, Q.B.; Khan, Y.; Hu, J.L. New families of third-order iterative methodsfor finding multiple roots. J. Appl. Math. 2014, 2014. [CrossRef]

14. Li, S.; Cheng, L.; Neta, B. Some fourth-order nonlinear solvers with closed formulae for multiple roots.Comput. Math. Appl. 2010, 59, 126–135. [CrossRef]

15. Ahmad, N.; Singh, V.P. Some New Three step Iterative methods for solving nonlinear equation usingSteffensen’s and Halley method. Br. J. Math. Comp. Sci. 2016, 19, 1–9. [CrossRef] [PubMed]

16. Neta, B. Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math.2010, 87, 1023–1031. [CrossRef]

17. Sharma, J.R.; Sharma, R. Modified Jarratt method for computing multiple roots. Appl. Math. Comput.2010, 217, 878–881. [CrossRef]

18. Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the multiple roots of nonlinearequations. J. Comput. Appl. Math. 2011, 235, 4199–4206. [CrossRef]

19. Kung, H.T.; Traub, J.F. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach.1974, 21, 643–651. [CrossRef]

20. Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. An eighth-order family of optimal multiple root findersand its dynamics. Numer. Algorithms 2018, 77, 1249–1272. [CrossRef]

21. Zafar, F.; Cordero, A.; Quratulain, R.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots ofnonlinear equations using free parameters. J. Math. Chem. 2017, 56, 1884–1091. [CrossRef]

22. Neta, B.; Chun, C.; Scott, M. On the development of iterative methods for multiple roots. Appl. Math. Comput.2013, 224, 358–361. [CrossRef]

23. Weerakoon, S.; Fernando, T.G.I. A variant of Newton’s method with accelerated third-order convergence.Appl. Math. Lett. 2000, 13, 87–93. [CrossRef]

24. Chicharro, F.I.; Cordero, A.; Torregrosa, J.R. Drawing dynamical and parameters planes of iterative familiesand methods. Sci. World J. 2013, 2013. [CrossRef] [PubMed]


148

mathematics

Article

An Efficient Conjugate Gradient Method for ConvexConstrained Monotone Nonlinear Equationswith Applications †

Auwal Bala Abubakar 1,2 , Poom Kumam 1,3,4,* , Hassan Mohammad 2

and Aliyu Muhammed Awwal 1,5

1 KMUTTFixed Point Research Laboratory, SCL 802 Fixed Point Laboratory, Science Laboratory Building,Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi(KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

2 Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria3 Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building,

King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod,Thrung Khru, Bangkok 10140, Thailand

4 Department of Medical Research, China Medical University Hospital, China Medical University,Taichung 40402, Taiwan

5 Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria* Correspondence: [email protected]† This project was supported by Petchra Pra Jom Klao Doctoral Academic Scholarship for Ph.D. Program at

KMUTT. Moreover, this project was partially supported by the Thailand Research Fund (TRF) and the KingMongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (GrantNo. RSA6080047).

Received: 29 June 2019; Accepted: 6 August 2019; Published: 21 August 2019

Abstract: This research paper proposes a derivative-free method for solving systems of nonlinearequations with closed and convex constraints, where the functions under consideration are continuousand monotone. Given an initial iterate, the process first generates a specific direction and then employsa line search strategy along the direction to calculate a new iterate. If the new iterate solves theproblem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set(constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descentcondition and the global convergence of the method is established under suitable assumptions.Finally, some numerical experiments were presented to show the performance of the proposedmethod in solving nonlinear equations and its application in image recovery problems.

Keywords: nonlinear monotone equations; conjugate gradient method; projection method;signal processing

MSC: 65K05; 90C52; 90C56; 92C55

1. Introduction

In this paper, we consider the following constrained nonlinear equation

F(x) = 0, subject to x ∈ Ψ, (1)

where F : Rn → Rn is continuous and monotone. The constraint set Ψ ⊂ Rn is nonempty, closedand convex.

Monotone equations appear in many applications [1–3], for example, the subproblems inthe generalized proximal algorithms with Bregman distance [4], reformulation of some �1-norm



regularized problems arising in compressive sensing [5] and variational inequality problems arealso converted into nonlinear monotone equations via fixed point maps or normal maps [6],(see References [7–9] for more examples). Among earliest methods for the case Ψ = Rn isthe hyperplane projection Newton method proposed by Solodov and Svaiter in Reference [10].Subsequently, many methods were proposed by different authors. Among the popular methodsare spectral gradient methods [11,12], quasi-Newton methods [13–15] and conjugate gradient methods(CG) [16,17].

To solve the constrained case (1), the work of Solodov and Svaiter was extended by Wang et al. [18]which also involves solving a linear system in each iteration but it was shown later by some authorsthat the computation of the linear system is not necessary. For examples, Xiao and Zhu [19] presented aCG method, which is a combination the well known CG-DESCENT method in Reference [20] with theprojection strategy by Solodov and Svaiter. Liu et al. [21] presented two CG method with sufficientlydescent directions. In Reference [22], a modified version of the method in Reference [19] was presentedby Liu and Li. The modification improves the numerical performance of the method in Reference [19].Another extension of the Dai and Kou (DK) CG method combined with the projection method tosolve (1) was proposed by Ding et al. in Reference [23]. Just recently, to popularize the Dai-Yuan (DY)CG method, Liu and Feng [24] modified the DY such that the direction will be sufficiently descent.A new hybrid spectral gradient projection method for solving convex constraints nonlinear monotoneequations was proposed by Awwal et al. in Reference [25]. The method is a convex combination of twodifferent positive spectral parameters together with the projection strategy. In addition, Abubakar et al.extended the method in Reference [17] to solve (1) and also solve some sparse signal recovery problems.

Inspired by some the above methods, we propose a descent conjugate gradient method to solveproblem (1). Under appropriate assumptions, the global convergence is established. Preliminarynumerical experiments were given to compare the proposed method with existing methods to solvenonlinear monotone equations and some signal and image reconstruction problems arising fromcompressive sensing.

The remaining part of this paper is organized as follows. In Section 2, we state the proposedalgorithm as well as its convergence analysis. Finally, Section 3 reports some numerical results to showthe performance of the proposed method in solving Equation (1), signal recovery problems and imagerestoration problems.

2. Algorithm: Motivation and Convergence Result

This section starts by defining the projection map together with some of its properties.

Definition 1. Let Ψ ⊂ Rn be a nonempty closed convex set. Then for any x ∈ Rn, its projection onto Ψ,denoted by PΨ(x), is defined by

PΨ(x) = arg min{‖x− y‖ : y ∈ Ψ}.

Moreover, PΨ is nonexpansive, That is,

‖PΨ(x)− PΨ(y)‖ ≤ ‖x− y‖, ∀x, y ∈ Rn. (2)

All through this article, we assume the followings

(G1) The mapping F is monotone, that is,

(F(x)− F(y))T(x− y) ≥ 0, ∀x, y ∈ Rn.

150


(G2) The mapping F is Lipschitz continuous, that is there exists a positive constant L such that

‖F(x)− F(y)‖ ≤ L‖x− y‖, ∀x, y ∈ Rn.

(G3) The solution set of (1), denoted by Ψ′, is nonempty.

An important property that methods for solving Equation (1) must possess is that the directiondk satisfy

F(xk)Tdk ≤ −c‖F(xk)‖2, (3)

where c > 0 is a constant. The inequality (3) is called sufficient descent property if F(x) is the gradientvector of a real valued function f : Rn → R.

In this paper, we propose the following search direction

dk =

{−F(xk), if k = 0,

−F(xk) + βkdk−1 − θkF(xk), if k ≥ 1,(4)

where

βk =‖F(xk)‖‖dk−1‖ (5)

and θk is determined such that Equation (3) is satisfied. It is easy to see that for k = 0, the equationholds with c = 1. Now for k ≥ 1,

F(xk)Tdk = −F(xk)

T F(xk) + F(xk)T ‖F(xk)‖‖dk−1‖ dk−1 − θkF(xk)

T F(xk)

= −‖F(xk)‖2 +‖F(xk)‖‖dk−1‖ F(xk)

Tdk−1 − θk‖F(xk)‖2

=−‖F(xk)‖2‖dk−1‖2 + ‖F(xk)‖‖dk−1‖F(xk)

Tdk−1 − θk‖F(xk)‖2‖dk−1‖2

‖dk−1‖2 .

(6)

Taking θk = 1 we haveF(xk)

Tdk ≤ −‖F(xk)‖2. (7)

Thus, the direction defined by (4) satisfy condition (3) ∀k where c = 1.To prove the global convergence of Algorithm 1, the following lemmas are needed.

Algorithm 1: (DCG)Step 0. Given an arbitrary initial point x0 ∈ Rn, parameters σ > 0, 0 < β < 1, Tol > 0 and set

k := 0.Step 1. If ‖F(xk)‖ ≤ Tol, stop, otherwise go to Step 2.Step 2. Compute dk using Equation (4).Step 3. Compute the step size αk = max{βi : i = 0, 1, 2, · · · } such that

− F(xk + αkdk)Tdk ≥ σαk‖F(xk + αkdk)‖‖dk‖2. (8)

Step 4. Set zk = xk + αkdk. If zk ∈ Ψ and ‖F(zk)‖ ≤ Tol, stop. Else compute

xk+1 = PΨ[xk − ζkF(zk)]

where

ζk =F(zk)

T(xk − zk)

‖F(zk)‖2 .

Step 5. Let k = k + 1 and go to Step 1.

151


Lemma 1. The direction defined by Equation (4) satisfies the sufficient descent property, that is, there existconstants c > 0 such that (3) holds.

Lemma 2. Suppose that assumptions (G1)–(G3) holds, then the sequences {xk} and {zk} generated byAlgorithm 1 (CGD) are bounded. Moreover, we have

limk→∞

‖xk − zk‖ = 0 (9)

andlimk→∞

‖xk+1 − xk‖ = 0. (10)

Proof. We will start by showing that the sequences {xk} and {zk} are bounded. Suppose x ∈ Ψ′,

then by monotonicity of F, we get

F(zk)T(xk − x) ≥ F(zk)

T(xk − zk). (11)

Also by definition of zk and the line search (8), we have

F(zk)T(xk − zk) ≥ σα2

k‖F(zk)‖‖dk‖2 ≥ 0. (12)

So, we have

‖xk+1 − x‖2 = ‖PΨ[xk − ζkF(zk)]− x‖2 ≤ ‖xk − ζkF(zk)− x‖2

= ‖xk − x‖2 − 2ζkF(zk)T(xk − x) + ‖ζF(zk)‖2

≤ ‖xk − x‖2 − 2ζkF(zk)T(xk − zk) + ‖ζF(zk)‖2

= ‖xk − x‖2 −(

F(zk)T(xk − zk)

‖F(zk)‖)2

≤ ‖xk − x‖2

(13)

Thus the sequence {‖xk − x‖} is non increasing and convergent and hence {xk} is bounded.Furthermore, from Equation (13), we have

‖xk+1 − x‖2 ≤ ‖xk − x‖2, (14)

and we can deduce recursively that

‖xk − x‖2 ≤ ‖x0 − x‖2, ∀k ≥ 0.

Then from Assumption (G2), we obtain

‖F(xk)‖ = ‖F(xk)− F(x)‖ ≤ L‖xk − x‖ ≤ L‖x0 − x‖.

If we let L‖x0 − x‖ = κ, then the sequence {F(xk)} is bounded, that is,

‖F(xk)‖ ≤ κ, ∀k ≥ 0. (15)

152


By the definition of zk, Equation (12), monotonicity of F and the Cauchy-Schwatz inequality,we get

σ‖xk − zk‖ = σ‖αkdk‖2

‖xk − zk‖ ≤F(zk)

T(xk − zk)

‖xk − zk‖ ≤ F(zk)T(xk − zk)

‖xk − zk‖ ≤ ‖F(xk)‖. (16)

The boundedness of the sequence {xk} together with Equations (15) and (16), implies the sequence{zk} is bounded.

Since {zk} is bounded, then for any x ∈ Ψ, the sequence {zk − x} is also bounded, that is, thereexists a positive constant ν > 0 such that

‖zk − x‖ ≤ ν.

This together with Assumption (G2) yields

‖F(zk)‖ = ‖F(zk)− F(x)‖ ≤ L‖zk − x‖ ≤ Lν.

Therefore, using Equation (13), we have

σ2

(Lν)2 ‖xk − zk‖4 ≤ ‖xk − x‖2 − ‖xk+1 − x‖2,

which implies

σ2

(Lν)2

∞

∑k=0‖xk − zk‖4 ≤

∞

∑k=0

(‖xk − x‖2 − ‖xk+1 − x‖2) ≤ ‖x0 − x‖ < ∞. (17)

Equation (17) implieslimk→∞

‖xk − zk‖ = 0.

However, using Equation (2), the definition of ζk and the Cauchy-Schwartz inequality, we have

‖xk+1 − xk‖ = ‖PΨ[xk − ζkF(zk)]− xk‖

≤ ‖xk − ζkF(zk)− xk‖

= ‖ζkF(zk)‖

= ‖xk − zk‖,

(18)

which yieldslimk→∞

‖xk+1 − xk‖ = 0.

Equation (9) and definition of zk implies that

limk→∞

αk‖dk‖ = 0. (19)

Lemma 3. Suppose dk is generated by Algorithm 1 (CGD), then there exist M > 0 such the ‖dk‖ ≤ M

153


Proof. By definition of dk and Equation (15)

‖dk‖ = ‖ − 2F(xk) +‖F(xk)‖‖dk−1‖ dk−1‖

≤ 2‖F(xk)‖+ ‖F(xk)‖‖dk−1‖ ‖dk−1‖

≤ 3‖F(xk)‖

≤ 3κ.

(20)

Letting M = 3κ, we have the desired result.

Theorem 1. Suppose that assumptions (G1)–(G3) hold and let the sequence {xk} be generated by Algorithm 1,then

lim infk→∞

‖F(xk)‖ = 0, (21)

Proof. To prove the Theorem, we consider two cases;Case 1

Suppose lim infk→∞

‖dk‖ = 0, we have lim infk→∞

‖F(xk)‖ = 0. Then by continuity of F, the sequence {xk} has

some accumulation point x such that F(x) = 0. Because {‖xk− x‖} converges and x is an accumulationpoint of {xk}, therefore {xk} converges to x.Case 2

Suppose lim infk→∞

‖dk‖ > 0, we have lim infk→∞

‖F(xk)‖ > 0. Then by (19), it holds that limk→∞

αk = 0.

Also from Equation (8),

−F(xk + βi−1dk)Tdk < σβi−1‖F(xk + βi−1dk)‖‖dk‖2

and the boundedness of {xk}, {dk}, we can choose a sub-sequence such that allowing k to go to infinityin the above inequality results

F(x)Td > 0. (22)

On the other hand, allowing k to approach ∞ in (7), implies

F(x)Td ≤ 0. (23)

(22) and (23) imply contradiction. Hence, lim infk→∞

‖F(xk)‖ > 0 is not true and the proof is complete.


This section gives the performance of the proposed method with existing methods such as PCGand PDY proposed in References [22,24], respectively, to solve monotone nonlinear equations using 9benchmark test problems. Furthermore Algorithm 1 is applied to restore a blurred image. All codeswere written in MATLAB R2018b and run on a PC with intel COREi5 processor with 4 GB of RAM andCPU 2.3 GHZ. All runs were stopped whenever ‖F(xk)‖ < 10−5.The parameters chosen for the existing algorithm are as follows:

PCG method: All parameters are chosen as in Reference [22].PDY method: All parameters are chosen as in Reference [24].Algorithm 1: We have tested several values of β ∈ (0, 1) and found that β = 0.7 gives the best

result. In addition, to implement most of the optimization algorithms, the parameter σ is chosen as

154


a very small number. Therefore, we chose β = 0.7 and σ = 0.0001 for the implementation of theproposed algorithm.

We test 9 different problems with dimensions ranging from n = 1000, 5000, 10, 000, 50, 000, 100, 000and 6 initial points: x1 = (0.1, 0.1, · · · , 1)T , x2 = (0.2, 0.2, · · · , 0.2)T , x3 = (0.5, 0.5, · · · , 0.5)T , x4 =

(1.2, 1.2, · · · , 1.2)T , x5 = (1.5, 1.5, · · · , 1.5)T , x6 = (2, 2, · · · , 2)T . In Tables 1–9, the number of iterations(ITER), number of function evaluations (FVAL), CPU time in seconds (TIME) and the norm at theapproximate solution (NORM) were reported. The symbol ‘−’ is used when the number of iterationsexceeds 1000 and/or the number of function evaluations exceeds 2000.

The test problems are listed below, where the function F is taken as F(x) =

( f1(x), f2(x), . . . , fn(x))T .

Problem 1 ([26]). Exponential Function.

f1(x) = ex1 − 1,

fi(x) = exi + xi − 1, for i = 2, 3, ..., n,

and Ψ = Rn+.

Problem 2 ([26]). Modified Logarithmic Function.

fi(x) = ln(xi + 1)− xin

, for i = 2, 3, ..., n,

and Ψ = {x ∈ Rn :

n

∑i=1

xi ≤ n, xi > −1, i = 1, 2, . . . , n}.

Problem 3 ([13]). Nonsmooth Function.

fi(x) = 2xi − sin |xi|, i = 1, 2, 3, ..., n,


n

∑i=1

xi ≤ n, xi ≥ 0, i = 1, 2, . . . , n}.

It is clear that Problem 3 is nonsmooth at x = 0.

Problem 4 ([26]). Strictly Convex Function I.

fi(x) = exi − 1, for i = 1, 2, ..., n,

and Ψ = Rn+.

Problem 5 ([26]). Strictly Convex Function II.

fi(x) =in

exi − 1, for i = 1, 2, ..., n,

and Ψ = Rn+.

Problem 6 ([27]). Tridiagonal Exponential Function

f1(x) = x1 − ecos(h(x1+x2)),

fi(x) = xi − ecos(h(xi−1+xi+xi+1)), for i = 2, ..., n− 1,

fn(x) = xn − ecos(h(xn−1+xn)),

h =1

n + 1and Ψ = R

n+.

155


Problem 7 ([28]). Nonsmooth Function

fi(x) = xi − sin |xi − 1|, i = 1, 2, 3, ..., n.


n

∑i=1

xi ≤ n, xi ≥ −1, i = 1, 2, . . . , n}.

Problem 8 ([23]). Penalty 1

ti =n

∑i=1

x2i , c = 10−5

fi(x) = 2c(xi − 1) + 4(ti − 0.25)xi, i = 1, 2, 3, ..., n.

and Ψ = Rn+.

Problem 9 ([29]). Semismooth Function

f1(x) = x1 + x31 − 10,

f2(x) = x2 − x3 + x32 + 1,

f3(x) = x2 + x3 + 2x33 − 3,

f4(x) = 2x34,

and Ψ = {x ∈ R4 :

4

∑i=1

xi ≤ 3, xi ≥ 0, i = 1, 2, 3, 4}.

In addition, we employ the performance profile developed in Reference [30] to obtain Figures 1–3,which is a helpful process of standardizing the comparison of methods. The measure of theperformance profile considered are; number of iterations, CPU time (in seconds) and number offunction evaluations. Figure 1 reveals that Algorithm 1 most performs better in terms of number ofiterations, as it solves and wins 90 percent of the problems with less number of iterations, while PCGand PDY solves and wins less than 10 percent. In Figure 2, Algorithm 1 performed a little less bysolving and winning over 80 percent of the problems with less CPU time as against PCG and PDY withsimilar performance of less than 10 percent of the problems considered. The translation of Figure 3 isidentical to Figure 1. Figure 4 is the plot of the decrease in residual norm against number of iterationson problem 9 with x4 as initial point. It shows the speed of the convergence of each algorithm using theconvergence tolerance 10−5, it can be observed that Algorithm 1 converges faster than PCG and PDY.

156


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164


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8

t

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Fig

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les

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the

num

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ions

.

165


0 1 2 3 4 5 6 7 8 9 10

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P(t

)

PCG

PDY

Algorithm 2.3

Figure 2. Performance profiles for the CPU time (in seconds).

0 1 2 3 4 5 6 7

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P(t

)

PDY

PCG

Algorithm 2.3

Figure 3. Performance profiles for the number of function evaluations.

10 15 20 25 30 35 40 45 50 55 60 65

ITERATION

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

RE

SID

UA

LS N

OR

M

Algorithm2.2PCGPDY

Figure 4. Convergence histories of Algorithm 1, PCG and PDY on Problem 9.

166


Applications in Compressive Sensing

There are many problems in signal processing and statistical inference involving finding sparsesolutions to ill-conditioned linear systems of equations. Among popular approach is minimizingan objective function which contains quadratic (�2) error term and a sparse �1−regularization term,that is,

minx

12‖y− Bx‖2

2 + η‖x‖1, (24)

where x ∈ Rn, y ∈ Rk is an observation, B ∈ Rk×n (k << n) is a linear operator, η is a non-negativeparameter, ‖x‖2 denotes the Euclidean norm of x and ‖x‖1 = ∑n

i=1 |xi| is the �1−norm of x. It is easyto see that problem (24) is a convex unconstrained minimization problem. Due to the fact that if theoriginal signal is sparse or approximately sparse in some orthogonal basis, problem (24) frequentlyappears in compressive sensing and hence an exact restoration can be produced by solving (24).

Iterative methods for solving (24) have been presented in many papers (see References [5,31–35]).The most popular method among these methods is the gradient based method and the earliest gradientprojection method for sparse reconstruction (GPRS) was proposed by Figueiredo et al. [5]. The first stepof the GPRS method is to express (24) as a quadratic problem using the following process. Let x ∈ Rn

and splitting it into its positive and negative parts. Then x can be formulated as

x = u− v, u ≥ 0, v ≥ 0,

where ui = (xi)+, vi = (−xi)+ for all i = 1, 2, ..., n and (.)+ = max{0, .}. By definition of �1-norm, wehave ‖x‖1 = eT

n u + eTn v, where en = (1, 1, ..., 1)T ∈ Rn. Now (24) can be written as

minu,v

12‖y− B(u− v)‖2

2 + ηeTn u + ηeT

n v, u ≥ 0, v ≥ 0, (25)

which is a bound-constrained quadratic program. However, from Reference [5], Equation (25) can bewritten in standard form as

minz

12

zT Dz + cTz, such that z ≥ 0, (26)

where z =

(uv

), c = ωe2n +

(−bb

), b = BTy, D =

(BT B −BT B−BT B BT B

).

Clearly, D is a positive semi-definite matrix, which implies that Equation (26) is a convexquadratic problem.

Xiao et al. [19] translated (26) into a linear variable inequality problem which is equivalentto a linear complementarity problem. Furthermore, it was noted that z is a solution of the linearcomplementarity problem if and only if it is a solution of the nonlinear equation:

F(z) = min{z, Dz + c} = 0. (27)

The function F is a vector-valued function and the “min” is interpreted as component-wise minimum.It was proved in References [36,37] that F(z) is continuous and monotone. Therefore problem (24) canbe translated into problem (1) and thus Algorithm 1 (DCG) can be applied to solve it.

In this experiment, we consider a simple compressive sensing possible situation, where our goalis to restore a blurred image. We use the following well-known gray test images; (P1) Cameraman,(P2) Lena, (P3) House and (P4) Peppers for the experiments. We use 4 different Gaussian blur kernalswith standard deviation σ to compare the robustness of DCG method with CGD method proposedin Reference [19]. CGD method is an extension of the well-known conjugate gradient method forunconstrained optimization CG-DESCENT [20] to solve the �1-norm regularized problems.

167


To access the performance of each algorithm tested with respect to metrics that indicate a betterquality of restoration, in Table 10 we reported the number of iterations, the objective function (ObjFun)value at the approximate solution, the mean of squared error (MSE) to the original image x,

MSE =1n‖x− x∗‖2,

where x∗ is the reconstructed image and the signal-to-noise-ratio (SNR) which is defined as

SNR = 20× log10( ‖x‖‖x− x‖

).

We also reported the structural similarity (SSIM) index that measure the similarity between the originalimage and the restored image [38]. The MATLAB implementation of the SSIM index can be obtainedat http://www.cns.nyu.edu/~lcv/ssim/.

Table 10. Efficiency comparison based on the value of the number of iterations (Iter), objective function(ObjFun) value, mean-square-error (MSE) and signal-to-noise-ratio (SNR) under different Pi (σ).

Image Iter ObjFun MSE SNR

DCG CGD DCG CGD DCG CGD DCG CGDP1(1E-8) 8 9 4.397 × 103 4.398 × 103 3.136 × 10−2 3.157 × 10−2 9.42 9.39P1(1E-1) 8 9 4.399 × 103 4.401 × 103 3.147 × 10−2 3.163 × 10−2 9.40 9.38P1(0.11) 11 8 4.428 × 103 4.432 × 103 3.229 × 10−2 3.232 × 10−2 9.29 9.29P1(0.25) 12 8 4.468 × 103 4.473 × 103 3.365 × 10−2 3.289 × 10−2 9.11 9.21

P1(1E-8) 9 9 4.555 × 103 4.556 × 103 3.287 × 10−2 3.3412 × 10−2 9.14 9.07P1(1E-1) 9 9 4.558 × 103 4.559 × 103 3.298 × 10−2 3.348 × 10−2 9.12 9.06P1(0.11) 12 12 4.588 × 103 4.591 × 103 3.416 × 10−2 3.446 × 10−2 8.97 8.93P1(0.25) 7 8 4.628 × 103 4.630 × 103 3.621 × 10−2 3.500 × 10−2 8.72 8.86

P1(1E-8) 9 9 5.179 × 103 5.179 × 103 3.209 × 10−2 3.3259 × 10−2 10.03 9.96P1(1E-1) 9 9 5.182 × 103 5.182 × 103 3.231 × 10−2 3.267 × 10−2 10.00 9.95P1(0.11) 7 9 5.209 × 103 5.209 × 103 3.436 × 10−2 3.344 × 10−2 9.73 9.85P1(0.25) 10 8 5.250 × 103 5.254 × 103 3.557 × 10−2 3.438 × 10−2 9.58 9.73

P1(1E-8) 9 9 4.388 × 103 4.389 × 103 3.299 × 10−2 3.335 × 10−2 9.03 8.99P1(1E-1) 9 9 4.391 × 103 4.393 × 103 3.308 × 10−2 3.340 × 10−2 9.02 8.98P1(0.11) 12 8 4.421 × 103 4.424 × 103 3.425 × 10−2 3.411 × 10−2 8.87 8.89P1(0.25) 7 8 4.461 × 103 4.463 × 103 3.621 × 10−2 3.483 × 10−2 8.63 8.80

The original, blurred and restored images by each of the algorithm are given in Figures 5–8.The figures demonstrate that both the two tested algorithm can restored the blurred images. It can beobserved from Table 10 and Figures 5–8 that Algorithm 1 (DCG) compete with the CGD algorithm,therefore, it can be used as an alternative to CGD for restoring blurred image.

168


Original Blurred

Recovered by CGD Recovered by DCG

Figure 5. The original image (top left), the blurred image (top right), the restored image by CGD(bottom left) with SNR = 20.05, SSIM = 0.83 and by DCG (bottom right) with SNR = 20.12, SSIM = 0.83.

Original Blurred



169


Original Blurred



Original Blurred



170


4. Conclusions

In this research article, we present a CG method which possesses the sufficient descent propertyfor solving constrained nonlinear monotone equations. The proposed method has the ability tosolve non-smooth equations as it does not require matrix storage and Jacobian information of thenonlinear equation under consideration. The sequence of iterates generated converge the solutionunder appropriate assumptions. Finally, we give some numerical examples to display the efficiency ofthe proposed method in terms of number of iterations, CPU time and number of function evaluationscompared with some related methods for solving convex constrained nonlinear monotone equationsand its application in image restoration problems.

Author Contributions: conceptualization, A.B.A.; methodology, A.B.A.; software, H.M.; validation, P.K. andA.M.A.; formal analysis, P.K. and H.M.; investigation, P.K. and A.M.A.; resources, P.K.; data curation, A.B.A. andH.M.; writing—original draft preparation, A.B.A.; writing—review and editing, H.M.; visualization, A.M.A.;supervision, P.K.; project administration, P.K.; funding acquisition, P.K.

Funding: Petchra Pra Jom Klao Doctoral Scholarship for Ph.D. program of King Mongkut’s University ofTechnology Thonburi (KMUTT) and Theoretical and Computational Science (TaCS) Center. Moreover, this projectwas partially supported by the Thailand Research Fund (TRF) and the King Mongkut’s University of TechnologyThonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047).

Acknowledgments: We thank Associate Professor Jin Kiu Liu for providing us with the access of the CGD-CSMATLAB codes. The authors acknowledge the financial support provided by King Mongkut’s University ofTechnology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. This project is supportedby the theoretical and computational science (TaCS) center under computational and applied science for smartresearch innovation (CLASSIC), Faculty of Science, KMUTT. The first author was supported by the “Petchra PraJom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi”.


References

1. Gu, B.; Sheng, V.S.; Tay, K.Y.; Romano, W.; Li, S. Incremental support vector learning for ordinal regression.IEEE Trans. Neural Netw. Learn. Syst. 2015, 26, 1403–1416. [CrossRef] [PubMed]

2. Li, J.; Li, X.; Yang, B.; Sun, X. Segmentation-based image copy-move forgery detection scheme. IEEE Trans.Inf. Forensics Secur. 2015, 10, 507–518.

3. Wen, X.; Shao, L.; Xue, Y.; Fang, W. A rapid learning algorithm for vehicle classification. Inf. Sci. 2015, 295, 395–406.[CrossRef]

4. Michael, S.V.; Alfredo, I.N. Newton-type methods with generalized distances for constrained optimization.Optimization 1997, 41, 257–278.

5. Figueiredo, M.A.T.; Nowak, R.D.; Wright, S.J. Gradient projection for sparse reconstruction: Application tocompressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 2007, 1, 586–597. [CrossRef]

6. Magnanti, T.L.; Perakis, G. Solving variational inequality and fixed point problems by line searches andpotential optimization. Math. Program. 2004, 101, 435–461. [CrossRef]

7. Pan, Z.; Zhang, Y.; Kwong, S. Efficient motion and disparity estimation optimization for low complexitymultiview video coding. IEEE Trans. Broadcast. 2015, 61, 166–176.

8. Xia, Z.; Wang, X.; Sun, X.; Wang, Q. A secure and dynamic multi-keyword ranked search scheme overencrypted cloud data. IEEE Trans. Parallel Distrib. Syst. 2016, 27, 340–352. [CrossRef]

9. Zheng, Y.; Jeon, B.; Xu, D.; Wu, Q.M.; Zhang, H. Image segmentation by generalized hierarchical fuzzyc-means algorithm. J. Intell. Fuzzy Syst. 2015, 28, 961–973.

10. Solodov, M.V.; Svaiter, B.F. A globally convergent inexact newton method for systems of monotone equations.In Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods; Springer: Dordrecht,The Netherlands, 1998; pp. 355–369.

11. Mohammad, H.; Abubakar, A.B. A positive spectral gradient-like method for nonlinear monotone equations.Bull. Comput. Appl. Math. 2017, 5, 99–115.

12. Zhang, L.; Zhou, W. Spectral gradient projection method for solving nonlinear monotone equations.J. Comput. Appl. Math. 2006, 196, 478–484. [CrossRef]

171


13. Zhou, W.J.; Li, D.H. A globally convergent BFGS method for nonlinear monotone equations without anymerit functions. Math. Comput. 2008, 77, 2231–2240. [CrossRef]

14. Zhou, W.; Li, D. Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 2007,25, 89–96.

15. Abubakar, A.B.; Waziria, M.Y. A matrix-free approach for solving systems of nonlinear equations. J. Mod.Methods Numer. Math. 2016, 7, 1–9. [CrossRef]

16. Abubakar, A.B.; Kumam, P. An improved three-term derivative-free method for solving nonlinear equations.Comput. Appl. Math. 2018, 37, 6760–6773. [CrossRef]

17. Abubakar, A.B.; Kumam, P.; Awwal, A.M. A descent dai-liao projection method for convex constrainednonlinear monotone equations with applications. Thai J. Math. 2018, 17, 128–152.

18. Wang, C.; Wang, Y.; Xu, C. A projection method for a system of nonlinear monotone equations with convexconstraints. Math. Methods Oper. Res. 2007, 66, 33–46. [CrossRef]

19. Xiao, Y.; Zhu, H. A conjugate gradient method to solve convex constrained monotone equations withapplications in compressive sensing. J. Math. Anal. Appl. 2013, 405, 310–319. [CrossRef]

20. Hager, W.; Zhang, H. A new conjugate gradient method with guaranteed descent and an efficient line search.SIAM J. Optim. 2005, 16, 170–192. [CrossRef]

21. Liu, S.-Y.; Huang, Y.-Y.; Jiao, H.-W. Sufficient descent conjugate gradient methods for solving convexconstrained nonlinear monotone equations. Abstr. Appl. Anal. 2014, 2014, 305643. [CrossRef]

22. Liu, J.K.; Li, S.J. A projection method for convex constrained monotone nonlinear equations with applications.Comput. Math. Appl. 2015, 70, 2442–2453. [CrossRef]

23. Ding, Y.; Xiao, Y.; Li, J. A class of conjugate gradient methods for convex constrained monotone equations.Optimization 2017, 66, 2309–2328. [CrossRef]

24. Liu, J.; Feng, Y. A derivative-free iterative method for nonlinear monotone equations with convex constraints.Numer. Algorithms 2018, 1–18. [CrossRef]

25. Muhammed, A.A.; Kumam, P.; Abubakar, A.B.; Wakili, A.; Pakkaranang, N. A new hybrid spectral gradientprojection method for monotone system of nonlinear equations with convex constraints. Thai J. Math. 2018,16, 125–147.

26. La Cruz, W.; Martínez, J.; Raydan, M. Spectral residual method without gradient information for solvinglarge-scale nonlinear systems of equations. Math. Comput. 2006, 75, 1429–1448. [CrossRef]

27. Bing, Y.; Lin, G. An efficient implementation of Merrill’s method for sparse or partially separable systems ofnonlinear equations. SIAM J. Optim. 1991, 1, 206–221. [CrossRef]

28. Yu, Z.; Lin, J.; Sun, J.; Xiao, Y.H.; Liu, L.Y.; Li, Z.H. Spectral gradient projection method for monotonenonlinear equations with convex constraints. Appl. Numer. Math. 2009, 59, 2416–2423. [CrossRef]

29. Yamashita, N.; Fukushima, M. Modified Newton methods for solving a semismooth reformulation ofmonotone complementarity problems. Math. Program. 1997, 76, 469–491. [CrossRef]

30. Dolan, E.D.; Moré, J.J. Benchmarking optimization software with performance profiles. Math. Program. 2002,91, 201–213. [CrossRef]

31. Figueiredo, M.A.T.; Nowak, R.D. An EM algorithm for wavelet-based image restoration. IEEE Trans.Image Process. 2003, 12, 906–916. [CrossRef]

32. Hale, E.T.; Yin, W.; Zhang, Y. A Fixed-Point Continuation Method for �1-Regularized Minimization withApplications to Compressed Sensing; CAAM TR07-07; Rice University: Houston, TX, USA, 2007; pp. 43–44.

33. Beck, A.; Teboulle, M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J.Imaging Sci. 2009, 2, 183–202. [CrossRef]

34. Van Den Berg, E.; Friedlander, M.P. Probing the pareto frontier for basis pursuit solutions. SIAM J.Sci. Comput. 2008, 31, 890–912. [CrossRef]

35. Birgin, E.G.; Martínez, J.M.; Raydan, M. Nonmonotone spectral projected gradient methods on convex sets.SIAM J. Optim. 2000, 10, 1196–1211. [CrossRef]

36. Xiao, Y.; Wang, Q.; Hu, Q. Non-smooth equations based method for �1-norm problems with applications tocompressed sensing. Nonlinear Anal. Theory Methods Appl. 2011, 74, 3570–3577. [CrossRef]

172


37. Pang, J.-S. Inexact Newton methods for the nonlinear complementarity problem. Math. Program. 1986, 36, 54–71.[CrossRef]

38. Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility tostructural similarity. IEEE Trans. Image Process. 2004, 13, 600–612. [CrossRef]


173

mathematics

Article

Advances in the Semilocal Convergence of Newton’sMethod with Real-World Applications

Ioannis K. Argyros 1, Á. Alberto Magreñán 2,*, Lara Orcos 3 and Íñigo Sarría 4

1 Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USA;[email protected]

2 Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain3 Departamento de Matemática Aplicada, Universidad Politècnica de València, 46022 València, Spain;

[email protected] Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño; Spain;

[email protected]* Correspondence: [email protected]

Received: 3 March 2019; Accepted: 21 March 2019; Published: 24 March 2019

Abstract: The aim of this paper is to present a new semi-local convergence analysis for Newton’smethod in a Banach space setting. The novelty of this paper is that by using more precise Lipschitzconstants than in earlier studies and our new idea of restricted convergence domains, we extendthe applicability of Newton’s method as follows: The convergence domain is extended; the errorestimates are tighter and the information on the location of the solution is at least as precise as before.These advantages are obtained using the same information as before, since new Lipschitz constantare tighter and special cases of the ones used before. Numerical examples and applications are usedto test favorable the theoretical results to earlier ones.

Keywords: Banach space; Newton’s method; semi-local convergence; Kantorovich hypothesis

1. Introduction

In this study we are concerned with the problem of approximating a locally unique solution z∗

of equationG(x) = 0, (1)

where G is a Fréchet-differentiable operator defined on a nonempty, open convex subset D of a Banachspace E1 with values in a Banach space E2.

Many problems in Computational disciplines such us Applied Mathematics, Optimization,Mathematical Biology, Chemistry, Economics, Medicine, Physics, Engineering and other disciplines canbe solved by means of finding the solutions of equations in a form like Equation (1) using MathematicalModelling [1–7]. The solutions of this kind of equations are rarely found in closed form. That is whymost solutions of these equations are given using iterative methods. A very important problem in thestudy of iterative procedures is the convergence region. In general this convergence region is small.Therefore, it is important to enlarge the convergence region without additional hypotheses.

The study of convergence of iterative algorithms is usually centered into two categories: Semi-localand local convergence analysis. The semi-local convergence is based on the information around aninitial point, to obtain conditions ensuring the convergence of theses algorithms while the localconvergence is based on the information around a solution to find estimates of the computed radii ofthe convergence balls.

Newton’s method defined for all n = 0, 1, 2, . . . by

zn+1 = zn − G′(zn)−1G(zn), (2)



is undoubtedly the most popular method for generating a sequence {zn} approximating z∗, where z0 isan initial point. There is a plethora of convergence results for Newton’s method [1–4,6,8–14]. We shallincrease the convergence region by finding a more precise domain where the iterates {zn} lie leading tosmaller Lipschitz constants which in turn lead to a tighter convergence analysis for Newton’s methodthan before. This technique can apply to improve the convergence domain of other iterative methodsin an analogous way.

Let us consider the conditions:There exist z0 ∈ Ω and η ≥ 0 such that

G′(z0)−1 ∈ L(E2, E1) and ‖G′(z0)

−1G(z0)‖ ≤ η;

There exists T ≥ 0 such that the Lipschitz condition

‖G′(z0)−1(G′(x)− G′(y))‖ ≤ T‖x− y‖

holds for all x, y ∈ Ω.Then, the sufficient convergence condition for Newton’s method is given by the famous for its

simplicity and clarity Kantorovich sufficient convergence criterion for Newton’s method

hK = 2Tη ≤ 1. (3)

Let us consider a motivational and academic example to show that this condition is not satisfied.Choose E1 = E2 = R, z0 = 1, p ∈ [0, 0.5), D = S(z0, 1− p) and define function G on D by

G(x) = z3 − p.

Then, we have T = 2(2− p). Then, the Kantorovich condition is not satisfied, since hK > 1 for allp ∈ (0, 0.5). We set IK = ∅ to be the set of point satisfying Equation (3). Hence, there is no guaranteethat Newton’s sequence starting at z0 converges to z∗ = 3

√p.

The rest of the paper is structured as follows: In Section 2 we present the semi-local convergenceanalysis of Newton’s method Equation (2). The numerical examples and applications are presented inSection 3 and the concluding Section 4.

2. Semi-Local Convergence Analysis

We need an auxiliary result on majorizing sequences for Newton’s method.

Lemma 1. Let H > 0, K > 0, L > 0, L0 > 0 and η > 0 be parameters. Suppose that:

h4 = L4η ≤ 1, (4)

where

L−14 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1L0 + H

, if b = LK + 2δL0(K− 2H) = 0

2−δ(L0 + H) +

√δ2(L0 + H)2 + δ(LK + 2δL0(K− 2H))

LK + 2δL0(K− 2H), if b > 0

−2δ(L0 + H) +

√δ2(L0 + H)2 + δ(LK + 2δL0(K− 2H))

LK + 2δL0(K− 2H), if b < 0

andδ =

2L

L +√

L2 + 8L0L.

175


holds. Then, scalar sequence {tn} given by

t0 = 0, t1 = η, t2 = t1 +K (t1−t0)

2

2 (1−H t1),

tn+2 = tn+1 +L (tn+1 − tn)2

2 (1− L0 tn+1)f or all n = 1, 2, · · · ,

(5)

is well defined, increasing, bounded from above by

t∗∗ = η +

(1 +

δ0

1− δ

)K η2

2 (1− H η)(6)

and converges to its unique least upper bound t∗ which satisfies

t2 ≤ t∗ ≤ t∗∗, (7)

where δ0 =L(t2 − t1)

2(1− L0t2). Moreover, the following estimates hold:

0 < tn+2 − tn+1 ≤ δ0 δn−1 K η2

2 (1− H η)f or all n = 1, 2, · · · (8)

and

t∗ − tn ≤ δ0 (t2 − t1)

1− δδn−2 f or all n = 2, 3, · · · . (9)

Proof. By induction, we show that

0 <L (tk+1 − tk)

2 (1− L0 tk+1)≤ δ (10)

holds for all k = 1, 2, · · · . Estimate Equation (10) is true for k = 1 by Equation (4). Then, we have byEquation (5)

0 < t3 − t2 ≤ δ0 (t2 − t1) =⇒ t3 ≤ t2 + δ0 (t2 − t1)

=⇒ t3 ≤ t2 + (1 + δ0) (t2 − t1)− (t2 − t1)

=⇒ t3 ≤ t1 +1−δ2

01−δ0

(t2 − t1) < t∗∗

and for m = 2, 3, · · ·

tm+2 ≤ tm+1 + δ0 δm−1 (t2 − t1)

≤ tm + δ0 δm−2 (t2 − t1) + δ0 δm−1 (t2 − t1)

≤ t1 + (1 + δ0 (1 + δ + · · ·+ δm−1)) (t2 − t1)

= t1 + (1 + δ01−δm

1−δ ) (t2 − t1) ≤ t∗∗.

176


Assume that Equation (10) holds for all natural integers n ≤ m. Then, we get byEquations (5) and (10) that

0 < tm+2 − tm+1 ≤ δ0 δm−1 (t2 − t1) ≤ δm (t2 − t1)

and

tm+2 ≤ t1 + (1 + δ01− δm

1− δ) (t2 − t1) ≤ t1 +

1− δm+1

1− δ(t2 − t1) < t∗∗.

Evidently estimate Equation (10) is true, if m is replaced by m + 1 provided that

L2(tm+2 − tm+1) ≤ δ (1− L0 tm+2)

orL2(tm+2 − tm+1) + δ L0 tm+2 − δ ≤ 0

orL2

δm (t2 − t1) + δ L0

(t1 +

1− δm+1

1− δ(t2 − t1)

)− δ ≤ 0. (11)

Estimate Equation (11) motivates us to define recurrent functions {ψk} on [0, 1) by

ψm(s) =L2(t2 − t1) tm+1 + s L0 (1 + s + t2 + · · ·+ tm) (t2 − t1)− (1− L0 t1) s.

We need a relationship between two consecutive functions ψk. We get that

ψm+1(s) = L2 (t2 − t1) tm+2 + s L0 (1 + s + t2 + · · ·+ tm+1) (t2 − t1)

−(1− L0 t1) s

= L2 (t2 − t1) tm+2 + s L0 (1 + s + t2 + · · ·+ tm+1) (t2 − t1)

−(1− L0 t1) s− L2 (t2 − t1) tm

−s L0 (1 + s + t2 + · · ·+ tm) (t2 − t1) + (1− L0 t1) s + ψk(s).

Therefore, we deduce that

ψm+1(s) = ψm(s) +12(2 L0 t2 + L s− L) tm (t2 − t1). (12)

Estimate Equation (11) is satisfied, if

ψm(δ) ≤ 0 holds for all m = 1, 2, · · · . (13)

Using Equation (12) we obtain that

ψm+1(δ) = ψm(δ) for all m = 1, 2, · · · .

Let us now define function ψ∞ on [0, 1) by

ψ∞(s) = limm→∞

ψm(s). (14)

177


Then, we have by Equation (14) and the choice of δ that

ψ∞(δ) = ψm(δ) for all m = 1, 2, · · · .

Hence, Equation (13) is satisfied, ifψ∞(δ) ≤ 0. (15)

Using Equation (11) we get that

ψ∞(δ) =

(L0

1− δ(t2 − t1) + L0 t1 − 1

)δ. (16)

It then, follows from Equations (2.1) and (2.13) that Equation (15) is satisfied. The induction isnow completed. Hence, sequence {tn} is increasing, bounded from above by t∗∗ given by Equation (6),and as such it converges to its unique least upper bound t∗ which satisfies Equation (7).

Let S(z, �) , S(z, �) stand, respectively for the open and closed ball in E1 with center z ∈ E1 and ofradius � > 0.

The conditions (A) for the semi-local convergence are:

(A1) G : D ⊂ E1 → E2 is Fréchet differentiable and there exist z0 ∈ D, η ≥ 0 such that G′(z0)−1 ∈

Ł(E2, E1) and‖G′(z0)

−1G(z0)‖ ≤ η.

(A2) There exists L0 > 0 such that for all x ∈ D

‖G′(z0)−1(G′(x)− G′(z0))‖ ≤ L0‖x− z0‖.

(A3) L0η < 1 and there exists L > 0 such that

‖G′(z0)−1(G′(x)− G′(y))‖ ≤ L‖x− y‖.

for all x, y ∈ D0 := S(z1,1L0− ‖G′(z0)

−1G(z0)‖) ∩ D.

(A4) There exists H > 0 such that

‖G′(z0)−1(G′(z1)− G′(z0))‖ ≤ H‖z1 − z0‖,

where z1 = z0 − G′(z0)−1G(z0).

(A5) There exists K > 0 such that for all θ ∈ [0, 1]

‖G′(z0)−1(G′(z0 + θ(z1 − z0))− G′(z0))‖ ≤ Kθ‖z1 − z0‖.

Notice that (A2) =⇒ (A3) =⇒ (A5) =⇒ (A4). Clearly, we have that

H ≤ K ≤ L0 (17)

and LL0

can be arbitrarily large [9]. It is worth noticing that (A3)–(A5) are not additional to (A2)

hypotheses, since in practice the computation of Lipschitz constant T requires the computation of theother constants as special cases.

Next, first we present a semi-local convergence result relating majorizing sequence {tn} withNewton’s method and hypotheses (A).

178


Theorem 1. Suppose that hypotheses (A), hypotheses of Lemma 1 and S(z0, t∗) ⊆ D hold, where t∗ is givenin Lemma 1. Then, sequence {zn} generated by Newton’s method is well defined, remains in S(z0, t∗) andconverges to a solution z∗ ∈ S(z0, t∗) of equation G(x) = 0. Moreover, the following estimates hold

‖zn+1 − zn‖ ≤ tn+1 − tn (18)

and‖zn − z∗‖ ≤ t∗ − tn f or all n = 0, 1, 2, · · · , (19)

where sequence {tn} is given in Lemma 1. Furthermore, if there exists R ≥ t∗ such that

S(z0, R) ⊆ D and L0 (t∗ + R) < 2,

then, the solution z∗ of equation G(x) = 0 is unique in S(z0, R).

Proof. We use mathematical induction to prove that

‖zk+1 − xk‖ ≤ tk+1 − tk (20)

andS(zk+1, t∗ − tk+1) ⊆ S(zk, t∗ − tk) for all k = 1, 2, · · · . (21)

Let z ∈ S(z1, t∗ − t1).Then, we obtain that

‖z− z0‖ ≤ ‖z− z1‖+ ‖z1 − z0‖ ≤ t∗ − t1 + t1 − t0 = t∗ − t0,

which implies z ∈ S(z0, t∗ − t0). Note also that

‖z1 − z0‖ = ‖G′(z0)−1 G(z0)‖ ≤ η = t1 − t0.

Hence, estimates Equations (20) and (21) hold for k = 0. Suppose these estimates hold for n ≤ k.Then, we have that

‖zk+1 − z0‖ ≤k+1

∑i=1‖zi − zi−1‖ ≤

k+1

∑i=1

(ti − ti−1) = tk+1 − t0 = tk+1

and‖zk + θ (zk+1 − zk)− z0‖ ≤ tk + θ (tk+1 − tk) ≤ t∗

for all θ ∈ (0, 1). Using Lemma 1 and the induction hypotheses, we get in turn that

‖G′(z0)−1(G′(zk+1)− G′(z0))‖ ≤ M‖xk+1 − z0‖ ≤ M(tk+1 − t0) ≤ Mtk+1 < 1, (22)

where

M =

{H if k = 0L0 if k = 1, 2, · · · .

It follows from Equation (22) and the Banach lemma on invertible operators that G′(zm+1)−1

exists and‖G′(zk+1)

−1 G′(z0)‖ ≤ (1− M ‖zk+1 − z0‖)−1 ≤ (1−M tk+1)−1. (23)

Using iteration of Newton’s method, we obtain the approximation

179


G(zk+1) = G(zk+1)− G(zk)− G′(zk) (zk+1 − zk)

=∫ 1

0(G′(zk + θ (zk+1 − zk))− G′(zm)) (zk+1 − zk) dθ.

(24)

Then, by Equation (24) we get in turn

‖G′(z0)−1 G(zk+1)‖

≤∫ 1

0‖G′(z0)

−1 (G′(zk + θ (zk+1 − zk))− G′(zk))‖ ‖zk+1 − zk‖ dθ

≤ M1

∫ 1

0‖θ (zk+1 − zk)‖ ‖zk+1 − zk‖ dθ ≤ M1

2(tk+1 − tk))

2,

(25)

where

M1 =

{K if k = 0L if k = 1, 2, · · · .

Moreover, by iteration of Newton’s method, Equations (23) and (25) and the induction hypotheseswe get that

‖zk+2 − zk+1‖ = ‖(G′(zk+1)−1 G′(z0)) (G′(z0)

−1 G(zk+1))‖

≤ ‖G′(zk+1)−1 G′(z0)‖ ‖G′(z0)

−1 G(zk+1)‖

≤M12 (tk+1−tk)

2

1−M tk+1= tk+2 − tk+1.

That is, we showed Equation (20) holds for all k ≥ 0. Furthermore, let z ∈ S(zk+2, t∗ − tk+2).Then, we have that

‖z− xk+1‖ ≤ ‖z− zk+2‖+ ‖zk+2 − zk+1‖

≤ t∗ − tk+2 + tk+2 − tk+1 = t∗ − tk+1.

That is, z ∈ S(zk+1, t∗ − tk+1). The induction for Equations (20) and (21) is now completed. Lemma1 implies that sequence {sn} is a complete sequence. It follows from Equations (20) and (21) that {zn}is also a complete sequence in a Banach space E1 and as such it converges to some z∗ ∈ S(z0, t∗) (sinceS(z0, t∗) is a closed set). By letting k −→ ∞ in Equation (25) we get G(∗) = 0. Estimate Equation (19) isobtained from Equation (18) (cf. [4,6,12]) by using standard majorization techniques. The proof for theuniqueness part has been given in [9].

The sufficient convergence criteria for Newton’s method using the conditions (A), constants L, L0

and η given in affine invariant form are:

• Kantorovich [6]hK = 2Tη ≤ 1. (26)

• Argyros [9]h1 = (L0 + T)η ≤ 1. (27)

• Argyros [3]

h2 =14

(T + 4L0 +

√T2 + 8L0T

)η ≤ 1 (28)

• Argyros [11]

h3 =14

(4L0 +

√L0T + 8L2

0 +√

L0T)

η ≤ 1 (29)

180


• Argyros [12]h4 = L4η ≤ 1,L4 = L4(T), δ = δ(T).

(30)

If H = K = L0 = L, then Equations (27)–(30) coincide with Equations (26). If L0 < T, then L < T

hK ≤ 1 ⇒ h1 ≤ 1 ⇒ h2 ≤ 1 ⇒ h3 ≤ 1 ⇒ h4 ≤ 1 ⇒ h5 ≤ 1,

but not vice versa. We also have that forL0

T→ 0 :

h1

hK→ 1

2,

h2

hK→ 1

4,

h2

h1→ 1

2

h3

hK→ 0,

h3

h1→ 0,

h3

h2→ 0

(31)

Conditions Equations (31) show by how many times (at most) the better condition improves theless better condition.

Remark 1. (a) The majorizing sequence {tn}, t∗, t∗∗ given in [12] under conditions (A) and Equation (29)is defined by

t0 = 0, t1 = η, t2 = t1 +L0(t1 − t0)

2

2(1− L0t1)

tn+2 = tn+1 +T(tn+1 − tn)2

2(1− L0tn+1), n = 1, 2, . . .

t∗ = limn→∞

tn ≤ t∗∗ = η +L0η2

2(1− δ)(1− L0η).

(32)

Using a simple inductive argument and Equation (32) we get for L1 < L that

tn < tn−1, n = 3, 4, . . . , (33)

tn+1 − tn < tn − tn−1, n = 2, 3, . . . , (34)

andt∗ ≤ t∗∗ (35)

Estimates for Equations (5)–(7) show the new error bounds are more precise than the old ones and theinformation on the location of the solution z∗ is at least as precise as already claimed in the abstract of thisstudy (see also the numerical examples). Clearly the new majorizing sequence {tn} is more precise thanthe corresponding ones associated with other conditions.

(b) Condition S(z0, t∗) ⊆ D can be replaced by S(z0, 1L0) (or D0). In this case condition (A2)

′ holds for allx, y ∈ S(z0, 1

L0) (or D0).

(c) If L0η ≤ 1, then, we have that z0 ∈ S(z1,1L0− ‖G′(z0)

−1G(z0)‖), since S(z1,1L0−

‖G′(z0)−1G(z0)‖) ⊆ S(z0,

1L0

).


Example 1. Returning back to the motivational example, we have L0 = 3− p.Conditions Equations (27)–(29) are satisfied, respectively for

p ∈ I1 := [0.494816242, 0.5),

181


p ∈ I2 := [0.450339002, 0.5)

andp ∈ I3 := [0.4271907643, 0.5).

We are now going to consider such an initial point which previous conditions cannot be satisfied but ournew criteria are satisfied. That is, the improvement that we get with our new weaker criteria.

We get that

H =5 + p

3,

K = 2,

L =2

3(3− p)(−2p2 + 5p + 6).

Using this values we obtain that condition Equation (4) is satisfied for p ∈ [0.0984119, 0.5), However,must also have that

L0η < 1

which is satisfied for p ∈ I4 := (0, 0.5]. That is, we must have p ∈ I4, so there exist numerous values of p forwhich the previous conditions cannot guarantee the convergence but our new ones can. Notice that we have

IK ⊆ I1 ⊆ I2 ⊆ I3 ⊆ I4

Hence, the interval of convergence cannot be improved further under these conditions. Notice that theconvergence criterion is even weaker than the corresponding one for the modified Newton’s method given in [11]by L0(η) < 0.5.

For example, we choose different values of p and we see in Table 1.

Table 1. Convergence of Newton’s method choosing z0 = 1, for different values of p.

p 0.41 0.43 0.45

z1 0.803333 0.810000 0.816667z2 0.747329 0.758463 0.769351z3 0.742922 0.754802 0.766321z4 0.742896 0.754784 0.766309z5 0.742896 0.754784 0.766309

Example 2. Consider E1 = E2 = A[0, 1]. Let D∗ = {x ∈ A[0, 1]; ‖x‖ ≤ R}, such that R > 0 and G definedon D∗ as

G(x)(u1) = x(u1)− f (u1)− λ∫ 1

0μ(u1, u2)x(u2)

3 du2, x ∈ C[0, 1], u1 ∈ [0, 1],

where f ∈ A[0, 1] is a given function, λ is a real constant and the kernel μ is the Green function. In this case,for all x ∈ D∗, G′(x) is a linear operator defined on D∗ by the following expression:

[G′(x)(v)](u1) = v(u2)− 3λ∫ 1

0μ(u1, u2)x(u2)

2v(u2) du2, v ∈ C[0, 1], u1 ∈ [0, 1].

If we choose z0(u1) = f (u1) = 1, it follows

‖I − G′(z0)‖ ≤ 3|λ|/8.

182


Hence, if|λ| < 8/3,

G′(z0)−1 is defined and

‖G′(z0)−1‖ ≤ 8

8− 3|λ| ,

‖G(z0)‖ ≤ |λ|8

,

η = ‖G′(z0)−1G(z0)‖ ≤ |λ|

8− 3|λ| .

Consider λ = 1.00, we getη = 0.2,

T = 3.8,

L0 = 2.6,

K = 2.28,

H = 1.28

andL = 1.38154 . . . .

By these values we conclude that conditions (26)–(29) are not satisfied, since

hK = 1.52 > 1,

h1 = 1.28 > 1,

h2 = 1.19343 . . . > 1,

h3 > 1.07704 . . . > 1,

but condition (2.27) and condition (4) are satisfied, since

h4 = 0.985779 . . . < 1

andh5 = 0.97017 . . . < 1.

Hence, Newton’s method converges by Theorem 1.

4. Application: Planck’s Radiation Law Problem

We consider the following problem [15] :

ϕ(λ) =8πcPλ−5

ecP

λBT−1(36)

which calculates the energy density within an isothermal blackbody. The maxima for ϕ occurs whendensity ϕ(λ). From (36), we get

ϕ′(λ) =(

8πcPλ−6

ecP

λBT−1

)(( cP

λBT )ecP

λBT−1

ecP

λkT−1− 5

)= 0, (37)

183


that is when( cP

λBT )ecP

λBT−1

ecP

λBT−1= 5. (38)

After using the change of variable x = cPλBT and reordering terms, we obtain

f (x) = e−x − 1 +x5

. (39)

As a consequence, we need to find the roots of Equation (39).We consider Ω = E(5, 1) ⊂ R and we obtain

η = 0.0348643 . . . ,

L0 = 0.0599067 . . . ,

K = 0.0354792 . . . ,

H = 0.0354792 . . .

andL = 0.094771 . . . .

So (A) are satisfied. Moreover, as b = 0.000906015 > 0, then

L4 = 10.0672 . . . ,

which satisfiesL4η = 0.350988 . . . < 1

and that means that conditions of Lemmal 1 are also satisfied. Finally, we obtain that

t∗ = 0.0348859 . . . .

Hence, Newton’s method converges to the solution x∗ = 4.965114231744276 . . . by Theorem 1.

Author Contributions: All authors have contributed in a similar way.

Funding: This research was supported in part by Programa de Apoyo a la investigación de lafundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14, by the projectMTM2014-52016-C2-1-P of the Spanish Ministry of Science and Innovation.


References

1. Amat, S.; Busquier, S.; Gutiérrez, J.M. Geometric constructions of iterative functions to solve nonlinearequations. J. Comput. Appl. Math. 2003, 157, 197–205. [CrossRef]

2. Amat, S.; Busquier, S. Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl.2007, 336, 243–261. [CrossRef]

3. Argyros, I.K. A semi-local convergence analysis for directional Newton methods. Math. Comput. 2011,80, 327–343. [CrossRef]

4. Argyros, I.K.; Magreñán, Á.A. Iterative Methods and Their Dynamics with Applications: A Contemporary Study;CRC Press: Bocaratón, FL, USA, 2017.

5. Farhane, N.; Boumhidi, I.; Boumhidi, J. Smart Algorithms to Control a Variable Speed Wind Turbine. Int. J.Interact. Multimed. Artif. Intell. 2017, 4, 88–95. [CrossRef]

6. Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Pergamon Press: Oxford, UK, 1982.7. Kaur, R.; Arora, S. Nature Inspired Range Based Wireless Sensor Node Localization Algorithms. Int. J.

Interact. Multimed. Artif. Intell. 2017, 4, 7–17. [CrossRef]

184


8. Amat, S.; Busquier, S.; Negra, M. Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim.2004, 25, 397–405. [CrossRef]

9. Argyros, I.K. On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Math. 2004, 169,315–332. [CrossRef]

10. Argyros, I.K.; González, D. Extending the applicability of Newton’s method for k-Fréchet differentiableoperators in Banach spaces. Appl. Math. Comput. 2014, 234, 167–178. [CrossRef]

11. Argyros, I.K.; Hilout, S. Weaker conditions for the convergence of Newton’s method. J. Complex. 2012,28, 364–387. [CrossRef]

12. Argyros, I.K.; Hilout, S. On an improved convergence analysis of Newton’s method. Appl. Math. Comput.2013, 225, 372–386. [CrossRef]

13. Ezquerro, J.A.; Hernández, M.A. How to improve the domain of parameters for Newton’s method.Appl. Math. Lett. 2015, 48, 91–101. [CrossRef]

14. Gutiérrez, J.M.; Magreñán, Á.A.; Romero, N. On the semi-local convergence of Newton-Kantorovich methodunder center-Lipschitz conditions. Appl. Math. Comput. 2013, 221, 79–88.

15. Divya, J. Families of Newton-like methods with fourth-order convergence. Int. J. Comput. Math. 2013,90, 1072–1082.


185

mathematics

Article

A Third Order Newton-Like Method andIts Applications

D. R. Sahu 1,*, Ravi P. Agarwal 2 and Vipin Kumar Singh 3

1 Department of Mathematics, Banaras Hindu University, Varanasi-221005, India2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA;

[email protected] Department of Mathematics, Banaras Hindu University, Varanasi-221005, India;


Received: 8 October 2018; Accepted: 7 December 2018; Published: 30 December 2018��

Abstract: In this paper, we design a new third order Newton-like method and establish itsconvergence theory for finding the approximate solutions of nonlinear operator equations in thesetting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-likemethod under the ω-continuity condition. Then we apply our approach to solve nonlinear fixed pointproblems and Fredholm integral equations, where the first derivative of an involved operator doesnot necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examplesare given, which compare the applicability of our convergence theory with the ones in the literature.

Keywords: nonlinear operator equation; Fréchet derivative; ω-continuity condition; Newton-likemethod; Frédholm integral equation

1. Introduction

Our purpose of this paper is to compute solution of nonlinear operator equation of the form

F(x) = 0, (1)

where F : D ⊂ X → Y is a nonlinear operator defined on an open convex subset D of a Banach spaceX with values into a Banach space Y.

A lot of challenging problems in physics, numerical analysis, engineering, and appliedmathematics are formulated in terms of finding roots of the equation of the form Equation (1).In order to solve such problems, we often use iterative methods. There are many iterative methodsavailable in literature. One of the central method for solving such problems is the Newton method [1,2]defined by

xn+1 = xn − (F′xn)−1F(xn) (2)

for each n ≥ 0, where F′x denotes the Fréchet derivative of F at point x ∈ D.The Newton method and the Newton-like method are attractive because it converges rapidly

from any sufficient initial guess. A number of researchers [3–20] have generalized and establishedlocal as well as semilocal convergence analysis of the Newton method Equation (2) under thefollowing conditions:

(a) Lipschitz condition: ‖F′x − F′y‖ ≤ K‖x− y‖ for all x, y ∈ D and for some K > 0;(b) Hölder Lipschitz condition: ‖F′x − F′y‖ ≤ K‖x − y‖p for all x, y ∈ D and for some p ∈ (0, 1]

and K > 0;



(c) ω-continuity condition: ‖F′x − F′y‖ ≤ ω(‖x − y‖) for all x, y ∈ D,where ω : [0, ∞) → [0, ∞) is anondecreasing and continuous function.

One can observe that the condition (c) is more general than the conditions (a) and (b). One canfind numerical examples where the Lipschitz condition (a) and the Hölder continuity condition (b) onthe first Fréchet derivative do not hold, but the ω-continuity condition (c) on first Fréchet derivativeholds (see Example 1, [21]).

On the other hand, many mathematical problems such as differential equations, integral equations,economics theory, game theory, variational inequalities, and optimization theory ([22,23]) can beformulated into the fixed point problem:

Find x ∈ C such that x = G(x), (3)

where G : C → X is an operator defined on a nonempty subset C of a metric space X. The easiestiterative method for constructing a sequence is Picard iterative method [24] which is given by

xn+1 = G(xn) (4)

for each n ≥ 0. The Banach contraction principle (see [1,22,23,25]) provides sufficient conditions forthe convergence of the iterative method Equation (4) to the fixed point of G. Banach spaces have moregeometrical stricture with respect to metric spaces. For study fixed points of nonlinear smooth operators,Banach space structure is required. More details of Banach space theory and fixed point theory ofnonlinear operators can be found in [1,22,23,26–28].

The Newton method and its variant [29,30] are also used to solve the fixed point problem ofthe form:

(I − G)(x) = 0, (5)

where I is the identity operator defined on X and G : D ⊂ X → X is a nonlinear Fréchet differentiableoperator defined on an open convex subset D of a Banach space X. For finding approximate solutionof the Equation (5), Bartle [31] used the Newton-like iterative method of the form

xn+1 = xn − (I − G′yn)−1

(I − G(xn)) (6)

for each n ≥ 0, where G′x is Fréchet derivative of G at point x ∈ D and {yn} is the sequence of arbitrarypoints in D which are sufficiently closed to the desired solution of the Equation (5). Bartle [31] hasdiscussed the convergence analysis of the form Equation (6) under the assumption that G is Fréchetdifferentiable at least at desired points and a modulus of continuity is known for G′ as a function of x.The Newton method Equation (2) and the modified Newton method are the special cases of the formEquation (6).

Following the idea of Bartle [31], Rall [32] introduced the following Stirling method for finding asolution of the fixed point problem Equation (5):{

yn = G(xn),xn+1 = xn − (I − G′yn)

−1(xn − G(xn))(7)

for each n ≥ 0. Many researchers [33–35] have studied the Stirling method Equation (7) and establishedlocal as well as semilocal convergence analysis of the Stirling-like method.

Recently, Parhi and Gupta [21,36] have discussed the semilocal convergence analysis of thefollowing Stirling-like iterative method for computing a solution of operator Equation (5):⎧⎪⎨⎪⎩

zn = G(xn),yn = xn − (I − G′zn)

−1(xn − G(xn)),xn+1 = yn − (I − G′zn)

−1(yn − G(yn))

(8)

187


for each n ≥ 0. More precisely, Parhi and Gupta [21] have studied the semilocal convergence analysisof Equation (8) for computing a solution of the operator Equation (5), where G : D ⊂ X → X is anonlinear Fréchet differentiable operator defined on an open convex subset D under the condition:

(Ω) ‖G′x‖ ≤ k for all x ∈ D and for some k ∈ (0, 13 ].

There are some nonlinear Fréchet differentiable operators G : D ⊂ X → X defined on an openconvex subset D which fail to satisfy the condition (Ω) (see Example 1). Therefore, ref. [21] (Theorem 1)is not applicable for such operators. So, there is the following natural question:

Problem 1. Is it possible to develop the Stirling-like iterative method for computing a solution of the operatorEquation (5), where the condition (Ω) does not hold?

The main purpose of this paper is to design a new Newton-like method for solving the operatorEquation (1) and provide an affirmative answer of the Problem 1. We prove our proposed Newton-likemethod has R-order of convergence at least 2p + 1 under the ω-continuity condition and it covers awide variety of iterative methods. We derive the Stirling-like iterative method for computing a solutionof the fixed point problem Equation (5), where (Ω) does not hold and hence it gives an affirmativeanswer to Question 1 and generalizes the results of Parhi and Gupta [21,36] in the context of thecondition (Ω).

In Section 2, we summarize some known concepts and results. In Section 3, we introduce a newNewton-like method for solving the operator Equation (1) and establish convergence theory of theproposed Newton-like method. In Section 4, we derive the Stirling-like iterative method from theproposed Newton-like method and establish a convergence theorem for computing a solution of thefixed point problem. Applications to Fredholm integral equations are also presented in Section 5,together with several numerical examples, which compare the applicability of our iterative techniquewith the ones in the literature.

2. Preliminary

In this section, we discuss some technical results. Throughout the paper, we denote B(X, Y)a collection of bounded linear operators from a Banach space X into a Banach space Y andB(X) = B(X, X). For some r > 0, Br[x] and Br(x) are the closed and open balls with center x andradius r, respectively, N0 = N ∪ {0} and Φ denote the collection of nonnegative, nondecreasing,continuous real valued functions defined on [0, ∞).

Lemma 1. (Rall [37] (p. 50)) Let L be a bounded linear operator on a Banach space X. Then L−1 exists if andonly if there is a bounded linear operator M in X such that M−1 exists and

‖M−L‖ <1

‖M−1‖ .

If L−1 exists, then we have

∥∥∥L−1∥∥∥ ≤ ∥∥M−1

∥∥1− ‖1−M−1L‖ ≤

∥∥M−1∥∥

1− ‖M−1‖ ‖M−L‖ .

Lemma 2. Let 0 < k ≤ 13 be a real number. Assume that q = 1

p+1 + kp for any p ∈ (0, 1] and thescalar equation

(1− kp(1 + qt)pt)p+1 −(

qptp

p + 1+ kp

)pqpt2p = 0

has a minimum positive root α. Then we have the following:

188


(1) q > k for all p ∈ (0, 1].(2) α ∈ (0, 1).

Proof. (1) This part is obvious. Indeed, we have

1p + 1

+ kp − 13=

2− p3(1 + p)

+ kp > 0

for all p ∈ (0, 1] and 0 < k ≤ 13 .

(2) Set

g(t) = (1− kp(1 + qt)pt)p+1 −(

qptp

p + 1+ kp

)pqpt2p. (9)

It is clear from the definition of g(t) that g(0) > 0, g(1) < 0 and g′(t) < 0 in (0, 1). Therefore, g(t)is decreasing in (0, 1) and hence the Equation (9) has a minimum positive root α ∈ (0, 1). This completesthe proof.

Lemma 3. Let b0 ∈ (0, α) be a number such that kp(1 + qb0)pb0 < 1, where k, p, α and q are same as in

Lemma 2. Define the real sequences {bn}, {θn} and {γn} by

bn+1 =

(qpbp

np+1 + kp

)pqpb2p

n

(1− kp(1 + qbn)pbn)p+1 bn, (10)

θn =

(qpbp

np+1 + kp

)qb2

n

1− kp(1 + qbn)pbn, γn =

11− kp(1 + qbn)pbn

(11)

for each n ∈ N0. Then we have the following:

(1)

(qpbp

0p+1 +kp

)p

qpb2p0

(1−kp(1+qb0)pb0)p+1 < 1.

(2) The sequence {bn} is decreasing, that is bn+1 ≤ bn for all n ∈ N0.(3) kp(1 + qbn)pbn < 1 for all n ∈ N0.(4) bn+1 ≤ ξ(2p+1)n

bn for all n ∈ N0.

(5) θn ≤ ξ(2p+1)n−1

p θ for all n ∈ N0, where θ0 = θ and ξ = γ0θp.

Proof. (1) Since the scalar equation g(t) = 0 defined by Equation (9) has a minimum positive rootα ∈ (0, 1) and g(t) is decreasing in (0, 1) with g(0) > 0 and g(1) < 0. Therefore, g(t) > 0 in theinterval (0, α) and hence (

qpbp0

p+1 + kp)p

qpb2p0

(1− kp(1 + qb0)pb0)p+1 < 1.

(2) From (1) and Equation (10), we have b1 ≤ b0. This shows that (2) is true for n = 0. Let j ≥ 0 bea fixed positive integer. Assume that (2) is true for n = 0, 1, 2, · · · , j. Now, using Equation (10),we have

bj+2 =

(qpbp

j+1p+1 + kp

)p

qpb2pj+1(

1− kp(1 + qbj+1)pbj+1)p+1 bj+1 ≤

(qpbp

jp+1 + kp

)p

qpb2pj(

1− kp(1 + qbj)pbn)p+1 bj = bj+1.

Thus (2) holds for n = j + 1. Therefore, by induction, (2) holds for all n ∈ N0.

189


(3) Since bn < bn−1 for each n = 1, 2, 3, · · · and kp(1 + qb0)pb0 < 1 for all p ∈ (0, 1], it follows that

kp(1 + qbn)pbn < kp(1 + qb0)

pb0 < 1.

(4) From (3), one can easily prove that the sequences {γn} and {θn} are well defined. Using Equations(10) and (11), one can easily observe that

bn+1 = γnθpnbn (12)

for each n ∈ N0. Put n = 0 and n = 1 in Equation (12), we have

b1 = γ0θpb0 = ξ(2p+1)0b0

and

b2 =

(qpbp

1p+1 + kp

)pqpb2p

1

(1− kp(1 + qb1)pb1)p+1 b1

≤

(qpbp

0p+1 + kp

)pqp(ξb0)

2p

(1− kp(1 + qb0)pb0)p+1 b1

≤ ξ2p

(qpbp

0p+1 + kp

)pqpb2p

0

(1− kp(1 + qb0)pb0)p+1 b1

= ξ2pγ0θpb1 = ξ2p+1b1.

Hence (4) holds for n = 0 and n = 1. Let j > 1 be a fixed integer. Assume that (4) holds for eachn = 0, 1, 2 · · · , j. From Equations (11) and (12), we have

bj+2 =

(qpbp

j+1p+1 + kp

)p

qpb2pj+1(

1− kp(1 + qbj+1)pbj+1)p+1 bj+1

≤

(qpbp

j+1p+1 + kp

)p

qp(ξ(2p+1)jbj)

2p

(1− kp(1 + qbj+1)pbj+1

)p+1 bj+1

≤ (ξ2p(2p+1)j)

(qpbp

jp+1 + kp

)p

qpb2pj(

1− kp(1 + qbj)pbj)p+1 bj+1

≤ ξ2p(2p+1)jξ2p(2p+1)j−1 · · · ξ2p(2p+1)ξ(2p+1)bj+1

= ξ(2p+1)j+1bj+1.

Thus (4) holds for n = j + 1. Therefore, by induction, (4) holds for all n ∈ N0.(5) From Equation (11) and (4), one can easily observe that

θ1 =

(qpbp

1p+1 + kp

)qb2

1

1− kp(1 + qb1)pb1≤

(qpbp

0p+1 + kp

)q(ξb0)

2

1− kp(1 + qb0)pb0≤ ξ

(2p+1)1−1p θ.

190


Hence (5) holds for n = 1. Let j > 1 be a fixed integer. Assume that (5) holds for eachn = 0, 1, 2 · · · , j. From Equation (11), we have

θj+1 =

(qpbp

j+1p+1 + kp

)qb2

j+1

1− kp(1 + qbj+1)pbj+1

≤

(qpbp

0p+1 + kp

)q

(ξ

(2p+1)j+1−12p b0

)2

1− kp(1 + qb0)pb0

= ξ(2p+1)j+1−1

p θ.

Thus (5) holds for n = j + 1. Therefore, by induction, (v) holds for all n ∈ N0. This completesthe proof.

3. Computation of a Solution of the Operator Equation (1)

Let X and Y be Banach spaces and D be a nonempty open convex subset of X. Let F : D ⊂ X → Ybe a nonlinear operator such that F is Fréchet differentiable at each point of D and let L ∈ B(Y, X) suchthat (I − LF)(D) ⊆ D. To solve the operator Equation (1), we introduce the Newton-like algorithmas follows:

Starting with x0 ∈ D and after xn ∈ D is defined, we define the next iterate xn+1 as follows:⎧⎪⎨⎪⎩zn = (I − LF)(xn),yn = (I − F′−1

zn F)(xn),xn+1 = (I − F′−1

zn F)(yn)

(13)

for each n ∈ N0.

If we take X = Y, F = I − G and L = I ∈ B(X) in Equation (13), then the iteration processEquation (13) reduces to the Stirling-like iteration process Equation (8).

Before proving the main result of the paper, we establish the following:

Proposition 1. Let D be a nonempty open convex subset of a Banach space X, F : D ⊂ X → Y be a Fréchetdifferentiable at each point of D with values in a Banach space Y and L ∈ B(Y, X) such that (I − LF)(D) ⊆ D.Let ω : [0, ∞)→ [0, ∞) be a nondecreasing and continuous real-valued function. Assume that F satisfies thefollowing conditions:

(1) ‖F′x − F′y‖ ≤ ω(‖x− y‖) for all x, y ∈ D;(2) ‖I − LF′x‖ ≤ c for all x ∈ D and for some c ∈ (0, ∞).

Define a mapping T : D → D byT(x) = (I − LF)(x) (14)

for all x ∈ D. Then we have‖I − F′−1

Tx F′Ty‖ ≤ ‖F′−1Tx ‖ω(c‖x− y‖)

for all x, y ∈ D.

Proof. For any x, y ∈ D, we have

191


‖I − F′−1Tx F′Ty‖ ≤ ‖F′−1

Tx ‖‖F′Tx − F′Ty‖≤ ‖F′−1

Tx ‖ω(‖Tx− Ty‖)= ‖F′−1

Tx ‖ω(‖x− y− L(F(x)− F(y))‖)= ‖F′−1

Tx ‖ω

(‖x− y− L

∫ 1

0F′y+t(x−y)(x− y)dt‖

)≤ ‖F′−1

Tx ‖ω

(∫ 1

0‖I − LF′y+t(x−y)‖dt‖x− y‖

)≤ ‖F′−1

Tx ‖ω(c‖x− y‖).

This completes the proof.

Now, we are ready to prove our main results for solving the problem Equation (1) in the frameworkof Banach spaces.

Theorem 1. Let D be a nonempty open convex subset of a Banach space X, F : D ⊂ X → Y a Fréchetdifferentiable at each point of D with values in a Banach space Y and L ∈ B(Y, X) such that (I − LF)(D) ⊆ D.Let x0 ∈ D be such that z0 = x0 − LF(x0) and F′−1

z0∈ B(Y, X) exist. Let ω ∈ Φ and let α be the solution of

the Equation (9). Assume that the following conditions hold:

(C1) ‖F′x − F′y‖ ≤ ω(‖x− y‖) for all x, y ∈ D;

(C2) ‖I − LF′x‖ ≤ k for all x ∈ D and for some k ∈ (0, 13 ];

(C3) ‖F′−1z0‖ ≤ β for some β > 0;

(C4) ‖F′−1z0

F(x0)‖ ≤ η for some η > 0;

(C5) ω(ts) ≤ tpω(s), s ∈ [0, ∞), t ∈ [0, 1] and p ∈ (0, 1];

(C6) b0 = βω(η) < α, q = 1p+1 + kp, θ =

(qpbp

0p+1 +kp

)qb2

0

1−kp(1+qb0)pb0

and Br[x0] ⊂ D, where r = 1+q1−θ η.

Then we have the following:

(1) The sequence {xn} generated by Equation (13) is well defined, remains in Br[x0] and satisfies thefollowing estimates: ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

‖yn−1 − zn−1‖ ≤ k‖yn−1 − xn−1‖,‖xn − yn−1‖ ≤ qbn−1‖yn−1 − xn−1‖,‖xn − xn−1‖ ≤ (1 + qbn−1)‖yn−1 − xn−1‖,F′−1

zn exists and ‖F′−1zn ‖ ≤ γn−1‖F′−1

zn−1‖,

‖yn − xn‖ ≤ θn−1‖yn−1 − xn−1‖ ≤ θn‖y0 − x0‖,‖F′−1

zn ‖ω(‖yn − xn‖) ≤ bn

(15)

for all n ∈ N, where zn, yn ∈ Br[x0], the sequences {bn}, {θn}, and {γn} are defined by Equations (10)and (11), respectively.

(2) The sequence {xn} converges to the solution x∗ ∈ Br[x0] of the Equation (1).(3) The priory error bounds on x∗ is given by:

‖xn − x∗‖ ≤ (1 + qb0)η

ξ1/2p2(

1− ξ(2p+1)n

p γ− 1

p0

)γ

n/p0

(ξ1/2p2

)(2p+1)n

for each n ∈ N0.(4) The sequence {xn} has R-order of convergence at least 2p + 1.

192


Proof. (1) First, we show that Equation (15) is true for n = 1. Since x0 ∈ D, y0 = x0 − F′−1z0

F(x0) iswell defined. Note that

‖y0 − x0‖ = ‖F′−1z0

F(x0)‖ ≤ η < r.

Hence y0 ∈ Br[x0]. Using Equation (13), we have

‖y0 − z0‖ = ‖ − F′−1z0

F(x0) + LF(x0)‖= ‖y0 − x0 − LF′z0

(y0 − x0)‖≤ ‖I − LF′z0

‖‖y0 − x0‖≤ k‖y0 − x0‖.

By Proposition 1 and (C2), we have

‖x1 − y0‖ = ‖F′−1z0

(F(y0)− F(x0)− F′z0(y0 − x0))‖

≤∫ 1

0‖F′−1

z0(F′x0+t(y0−x0)

− F′y0+ F′y0

− F′z0)‖‖y0 − x0‖dt

≤ β

[∫ 1

0‖(F′x0+t(y0−x0)

− F′y0)‖dt + ‖F′y0

− F′z0‖]‖y0 − x0‖

= β

[∫ 1

0ω((1− t)‖y0 − x0‖)dt + ω(‖y0 − z0‖)

]‖y0 − x0‖

= β

[∫ 1

0(1− t)pω(‖y0 − x0‖)dt + ω(k‖y0 − x0‖)

]‖y0 − x0‖

= β

[∫ 1

0(1− t)pω(‖y0 − x0‖)dt + kpω(‖y0 − x0‖)

]‖y0 − x0‖

≤ β

[1

p + 1+ kp

]ω(‖y0 − x0‖)‖y0 − x0‖

≤ qβω(η)‖y0 − x0‖ ≤ qb0‖y0 − x0‖.

Thus we have

‖x1 − x0‖ ≤ ‖x1 − y0‖+ ‖y0 − x0‖ ≤ qb0‖y0 − x0‖+ ‖y0 − x0‖≤ (1 + qb0)‖y0 − x0‖ < r,

(16)

which shows that x1 ∈ Br[x0]. Note that z1 = (I − LF)(x1) ∈ D. Using Proposition 1 and (C3)–(C5),we have

‖I − F′−1z0

F′z1‖ ≤ ‖F′−1

z0‖ω(k‖x1 − x0‖)

≤ βω(k(1 + qb0)‖y0 − x0‖)≤ βkp(1 + qb0)

pω(‖y0 − x0‖)≤ kp(1 + qb0)

pβω(η)

≤ (k(1 + qb0))pb0 < 1.

Therefore, by Lemma 1, F′−1z1

exists and

‖F′−1z1‖ ≤ ‖F′−1

z0‖

1− (k(1 + qb0))pb0= γ0‖F′−1

z0‖. (17)

Subsequently, we have

193


‖y1 − x1‖ = ‖F′−1z1

F(x1)‖= ‖F′−1

z1(F(x1)− F(y0)− F′z0

(x1 − y0))‖≤ ‖F′−1

z1‖[∫ 1

0 ‖(F′y0+t(x1−y0)− F′y0

)‖dt + ‖F′y0− F′z0

‖]‖x1 − y0‖

≤ ‖F′−1z1‖[

1p+1 ω(‖x1 − y0‖) + ω(k‖y0 − x0‖)

]‖x1 − y0‖

≤ ‖F′−1z1‖[

1p+1 ω(qb0‖x0 − y0‖) + kpω(‖y0 − x0‖)

]qb0‖x0 − y0‖

≤ ‖F′−1z1‖[

qpbp0

p+1 ω(‖y0 − x0‖) + kpω(‖y0 − x0‖)]

qb0‖y0 − x0‖

≤ γ0

[qpbp

0p+1 + kp

]βω(η)qb0‖y0 − x0‖

≤

(qpbp

0p+1 +kp

)qb2

0

1−(k(1+qb0))pb0‖y0 − x0‖

≤ θ‖y0 − x0‖.

(18)

From Equations (16) and (18), we have

‖y1 − x0‖ ≤ ‖y1 − x1‖+ ‖x1 − x0‖≤ θ‖y0 − x0‖+ (1 + qb0)‖y0 − x0‖≤ (1 + qb0)θ‖y0 − x0‖+ (1 + qb0)‖y0 − x0‖≤ (1 + qb0)(1 + θ)η < r

and

‖z1 − x0| ≤ ‖z1 − y1‖+ ‖y1 − x1‖+ ‖x1 − x0‖≤ (1 + k)‖y1 − x1‖+ (1 + qb0)‖y0 − x0‖≤ (1 + q)θη + (1 + q)η

= (1 + q)(1 + θ)η < r.

This shows that z1, y1 ∈ Br[x0]. From Equations (17) and (18), we get

‖F′−1z1‖ω(‖y1 − x1‖) ≤ γ0‖F′−1

z0‖ω(θ‖y0 − x0‖)

≤ γ0θpβω(η)

≤ γ0θpb0 = b1.

Thus we see that Equation (15) holds for n = 1.Let j > 1 be a fixed integer. Assume that Equation (15) is true for n = 1, 2, · · · , j. Since xj ∈ Br[x0],

it follows zj = (I − LF)(xj) ∈ D. Using (C3), (C4), Equations (13) and (15), we have

‖yj − zj‖ = ‖LF(xj)− F′−1zj

F(xj)‖ = ‖(L− F′−1zj

)F(xj)‖= ‖(L− F′−1

zj)F′zj

(xj − yj)‖≤ ‖I − LF′zj

‖‖yj − xj‖≤ k‖yj − xj‖.

(19)

Using Equations (13) and (19), we have

194


‖xj+1 − yj‖ = ‖F′−1zj

F(yj)‖≤ ‖F′−1

zj‖‖F(yj)− F(xj)− F′zj

(yj − xj)‖≤ ‖F′−1

zj‖[∫ 1

0 ‖F′xj+t(yj−xj)− F′zj

‖dt]‖yj − xj‖

≤ ‖F′−1zj‖[∫ 1

0 ‖F′xj+t(yj−xj)− F′yj

‖dt + ‖F′yj− F′zj

‖]‖yj − xj‖

= ‖F′−1zj‖[∫ 1

0 ω(xj + t(yj − xj)− yj)dt + ω(‖yj − zj‖)]‖yj − xj‖

≤ ‖F′−1zj‖[∫ 1

0 ω((1− t)‖yj − xj‖)dt + ω(k‖yj − xj‖)]‖yj − xj‖

≤ ‖F′−1zj‖[∫ 1

0 ((1− t)p + kp)ω(‖yj − xj‖)dt]‖yj − xj‖

≤ ‖F′−1zj‖[

1p+1 + kp

]ω(‖yj − xj‖)‖yj − xj‖

= q‖F′−1zj‖ω(‖yj − xj‖)‖yj − xj‖

= qbj‖yj − xj‖.

(20)

From Equation (20), we have

‖xj+1 − xj‖ ≤ ‖xj+1 − yj‖+ ‖yj − xj‖≤ qbj‖yj − xj‖+ ‖yj − xj‖≤ (1 + qbj)‖yj − xj‖.

(21)

Using Equations (20) and (21) and the triangular inequality, we have

‖xj+1 − x0‖ ≤j

∑s=0‖xs+1 − xs‖

≤j

∑s=0

(1 + qbs)‖ys − xs‖

≤j

∑s=0

(1 + qb0)θs‖y0 − x0‖

≤ (1 + qb0)1− θ j+1

1− θη

≤ (1 + q)η1− θ

= r,

which implies that xk+1 ∈ Br[x0]. Again, by using Proposition 1, (C2), (C5), and Equation (21),we have

‖I − F′−1zj

F′zj+1‖ ≤ ‖F′−1

zj‖ω(k‖xj+1 − xj‖)

≤ ‖F′−1zj‖kp(1 + qbj)

pω(‖yj − xj‖)≤ kp(1 + qbj)

pbj < 1.

Therefore, by Lemma 1, F′−1zj+1

exists and

‖F′−1zj+1‖ ≤

‖F′−1zj‖

1− kp(1 + qbj)pbj= γj‖F′−1

zj‖.

Using Equations (13), (C2), and (21), we have

195


‖yj+1 − xj+1‖ = ‖F′−1zj+1

F(xj+1)‖= ‖F′−1

zj+1(F(xj+1)− F(yj)− F′zj

(xj+1 − yj))‖

≤ ‖F′−1zj+1‖[∫ 1

0‖F′yj+t(xj+1−yj)

− F′yj‖dt + ‖F′yj

− F′zj‖]‖xj+1 − yj‖

≤ ‖F′−1zj+1‖[∫ 1

0ω(t‖xj+1 − yj‖)dt + ω(‖yj − zj‖)

]‖xj+1 − yj‖

≤ ‖F′−1zj+1‖[∫ 1

0ω(tqbj‖yj − xj‖)dt + ω(k‖yj − xj‖)

]qbj‖yj − xj‖

≤ γj‖F′−1zj‖[

qpbpj

p + 1ω(‖yj − xj‖) + kpω(‖yj − xj‖)

]qbj‖yj − xj‖

= γj

[qpbp

j

p + 1+ kp

]‖F′−1

zj‖ω(‖yj − xj‖)qbj‖yj − xj‖

≤ γj

[qpbp

j

p + 1+ kp

]qb2

j ‖yj − xj‖

≤ θj‖yj − xj‖ ≤ θ j+1‖y0 − x0‖,

‖yj+1 − x0‖ ≤ ‖yj+1 − xj+1‖+ ‖xj+1 − x0‖

≤ θ j+1‖y0 − x0‖+j

∑s=0‖xs+1 − xs‖

≤ θ j+1‖y0 − x0‖+j

∑s=0

(1 + qb0)θs‖y0 − x0‖

≤ (1 + qb0)j+1

∑s=0

θsη

≤ (1 + q)η1− θ

= r

and

‖zj+1 − x0‖ ≤ ‖zj+1 − yj+1‖+ ‖yj+1 − xj+1‖+ ‖xj+1 − x0‖

≤ (1 + k)‖yj+1 − xj+1‖+j

∑s=0

(1 + qb0)θsη

≤ (1 + q)θ j+1η +j

∑s=0

(1 + q)θsη

≤j+1

∑s=0

(1 + q)θsη < r

which implies that zj+1, yj+1 ∈ Br(x0). Also, we have

‖F′−1zj+1‖ω(‖yj+1 − xj+1‖) ≤ γj‖F′−1

zj‖ω(θj‖yj − xj‖)

≤ γjθpj ‖F′−1

zj‖ω(‖yj − xj‖)

≤ γjθpj bj = bj+1.

Hence we conclude that Equation (15) is true for n = j + 1. Therefore, by induction, Equation (15)is true for all n ∈ N0.

196


(2) First, we show that the sequence {xn} is a Cauchy sequence. For this, letting m, n ∈ N0 andusing Lemma 3, we have

‖xm+n − xn‖ ≤m+n−1

∑j=n

‖xj+1 − xj‖

≤m+n−1

∑j=n

(1 + qbj)‖yj − xj‖

≤ (1 + qb0)m+n−1

∑j=n

j−1

∏i=0

θi‖y0 − x0‖

≤ (1 + qb0)m+n−1

∑j=n

j−1

∏i=0

ξ(2p+1)i−1

p θ‖y0 − x0‖

≤ (1 + qb0)m+n−1

∑j=n

j−1

∏i=0

ξ(2p+1)i

p γ− 1

p0 ‖y0 − x0‖

= (1 + qb0)m+n−1

∑j=n

ξ

j−1

∑i=0

(2p + 1)i

pγ− 1

p0 ‖y0 − x0‖

≤ (1 + qb0)

(m+n−1

∑j=n

ξ(2p+1)j−1

2p2 γ− j

p0

)‖y0 − x0‖.

By Bernoulli’s inequality, for each j ≥ 0 and y > −1, we have (1 + y)j ≥ 1 + jy. Hence we have

‖xm+n − xn‖≤ (1 + qb0)ξ

− 12p2 γ

− np

0

(ξ

(2p+1)n

2p2 + ξ(2p+1)n(2p+1)

2p2 γ− 1

p0 + · · ·+ ξ

(2p+1)n(2p+1)m−1

2p2 γ− (m−1)

p0

)η

≤ (1 + qb0)ξ− 1

2p2 γ− n

p0

(ξ

(2p+1)n

2p2 + ξ(2p+1)n(1+2p)

2p2 γ− 1

p0 + · · ·+ ξ

(2p+1)n(1+2(m−1)p)2p2 γ

− (m−1)p

0

)η

= (1 + qb0)ξ− 1

2p2 γ− n

p0

⎛⎝ξ(2p+1)n

2p2 + ξ(2p+1)n

(1

2p2 +1p

)γ− 1

p0 + · · ·+ ξ

(2p+1)n(

12p2 +

m−1p

)γ− (m−1)

p0

⎞⎠ η

= (1 + qb0)ξ(2p+1)n−1

2p2 γ− n

p0

(1 +

(ξ(2p+1)n

γ−10

) 1p+ · · ·+

(ξ(2p+1)n

γ−10

)m−1p)

η

= (1 + qb0)ξ(2p+1)n−1

2p2 γ− n

p0

⎛⎝ 1−(

ξ(2p+1)n γ−10

)mp

1−(ξ(2p+1)n γ−10 )

1p

⎞⎠ η.

(22)

Since the sequence {xn} is a Cauchy sequence and hence it converges to some point x∗ ∈ Br[x0].From Equations (13), (C2), and (15), we have

‖LF(xn)‖ ≤ ‖zn − yn‖+ ‖yn − xn‖≤ k‖yn − xn‖+ ‖yn − xn‖≤ (1 + k)θnη.

197


Taking the limit as n → ∞ and using the continuity of F and the linearity of L, we have

F(x∗) = 0.

(3) Taking the limit as m → ∞ in Equation (22), we have

‖x∗ − xn‖ ≤ (1 + qb0)η

ξ1/2p2(

1− ξ(2p+1)n

p γ− 1

p0

)γ

n/p0

(ξ1/2p2

)(2p+1)n

(23)

for each n ∈ N0.(4) Here we prove

‖xn+1 − x∗‖‖xn − x∗‖2p+1 ≤ K

for all n ∈ N0 and for some K > 0. One can easily observe that there exists n0 > 0 such that

‖xn − x∗‖ < 1 (24)

whenever n ≥ n0. Using Equations (13) and (24), we have

‖zn − x∗‖ = ‖xn − x∗ − LF(xn)‖= ‖xn − x∗ − L(F(xn)− F(x∗))‖= ‖xn − x∗ − L

∫ 1

0F′x∗+t(xn−x∗)(xn − x∗)dt‖

≤∫ 1

0‖I − LF′x∗+t(xn−x∗)‖‖xn − x∗‖dt

≤ k‖xn − x∗‖

and‖yn − x∗‖ = ‖xn − x∗ − F′−1

zn F(xn)‖= ‖F′−1

zn [F′zn(xn − x∗)− F(xn)]‖≤ ‖F′−1

zn ‖‖F(xn)− x∗ − F′zn(xn − x∗)‖= ‖F′−1

zn ‖‖ ∫ 10 (F′x∗+t(xn−x∗) − F′zn)(xn − x∗)‖dt

≤ ‖F′−1zn ‖ ∫ 1

0 ‖F′x∗+t(xn−x∗) − F′zn‖‖xn − x∗‖dt

≤ ‖F′−1zn ‖ ∫ 1

0 (‖F′x∗+t(xn−x∗) − F′x∗‖+ ‖F′x∗ − F′zn‖)‖xn − x∗‖dt

≤ ‖F′−1zn ‖ ∫ 1

0 (ω(t‖xn − x∗‖) + ω(‖zn − x∗‖))‖xn − x∗‖dt

≤ ‖F′−1zn ‖ ∫ 1

0 (tp‖xn − x∗‖pω(1) + ω(k‖xn − x∗‖))‖xn − x∗‖dt

≤ ‖F′−1zn ‖

(1

p+1 + kp)

ω(1)‖xn − x∗‖p+1

= ‖F′−1zn ‖qω(1)‖xn − x∗‖p+1.

(25)

Using Equations (13), (24) and (25), we have

198


‖xn+1 − x∗‖= ‖yn − F′−1

zn F(yn)− x∗‖≤ ‖F′−1

zn ‖‖F(yn)− F(x∗)− F′zn(yn − x∗)‖= ‖F′−1

zn ‖‖∫ 1

0(F′x∗+t(yn−x∗) − F′zn)(yn − x∗)‖dt

≤ ‖F′−1zn ‖

∫ 1

0‖F′x∗+t(yn−x∗) − F′zn‖‖yn − x∗‖dt

= ‖F′−1zn ‖

∫ 1

0(‖F′x∗+t(yn−x∗) − F′x∗‖+ ‖F′x∗ − F′zn‖)‖yn − x∗‖dt

≤ ‖F′−1zn ‖

∫ 1

0(ω(t‖yn − x∗‖) + ω(k‖xn − x∗‖))‖yn − x∗‖dt

= ‖F′−1zn ‖

∫ 1

0

(tpω

(‖F′−1

zn ‖qω(1)‖xn − x∗‖p+1)+ ω(k‖xn − x∗‖)

)dt

×‖F′−1zn ‖qω(1)‖xn − x∗‖p+1

≤ ‖F′−1zn ‖2

∫ 1

0

(tp‖xn − x∗‖p(p+1)ω

(‖F′−1

zn ‖qω(1))+ kp‖xn − x∗‖pω(1)

)dt

×qω(1)‖xn − x∗‖p+1

= ‖F′−1zn ‖2

(‖xn − x∗‖p2ω(‖F′−1

zn ‖qω(1))

p + 1+ kp

)qω(1)‖xn − x∗‖2p+1

= Kn‖xn − x∗‖2p+1,

where

Kn = ‖F′−1zn ‖2

(‖xn − x∗‖p2ω(‖F′−1

zn ‖qω(1))

p + 1+ kp

)qω(1).

Let ‖F′−1x∗ ‖ ≤ d and 0 < d < ω(σ)−1, where σ > 0. Then, for all x ∈ Bσ(x∗), we have

‖I − F′−1x∗ F′x‖ ≤ ‖F′−1

x∗ ‖‖F′x∗ − F′x‖ ≤ dω(σ) < 1

and so, by Lemma 1, we have

‖F′−1x ‖ ≤ d

1− dω(σ):= λ.

Since xn → x∗ and zn → x∗ as n → ∞, there exists a positive integer N0 such that

‖F′−1zn ‖ ≤ d

1− dω(σ)

for all n ≥ N0. Thus, for all n ≥ N0, one can easily observe that

Kn ≤ λ2(

σp2ω (λqω(1))

p + 1+ kp

)qω(1) = K.

This shows that the R-order of convergence at least (2p + 1). This completes the proof.

4. Applications

4.1. Fixed Points of Smooth Operators

Let X be a Banach spaces and D be a nonempty open convex subset of X. Let G : D ⊂ X → X bea nonlinear operator such that D is Fréchet differentiable at each point of D and let L ∈ B(X, X) suchthat (I − L(I − G))(D) ⊆ D. For F = I − G, the Newton-like algorithm Equation (13) reduces to thefollowing Stirling-like method for computing fixed point of the operator G:

199


Starting with x0 ∈ D and after xn ∈ D is defined, we define the next iterate xn+1 as follows:⎧⎪⎨⎪⎩zn = (I − L(I − G))(xn),yn = (I − (I − G′zn)

−1(I − G))(xn),xn+1 = (I − (I − G′zn)

−1(I − G))(yn)

(26)

for each n ∈ N0.For the choice of X = Y and F = I − G, Theorem 1 reduces to the following:

Theorem 2. Let D be a nonempty open convex subset of a Banach space X, G : D → X be a Fréchetdifferentiable at each point of D with values into itself. Let L ∈ B(X) be such that (I − L(I − G))(D) ⊆ D.Let x0 ∈ D be such that z0 = x0 − L(x0 − G(x0)) and let (I − G′z0

)−1 ∈ B(X) exist. Let ω ∈ Φ and α be asolution of the Equation (9). Assume that the conditions (C5)–(C6) and the following conditions hold:

(C7) ‖(I − G′z0)−1‖ ≤ β for some β > 0;

(C8) ‖(I − G′z0)−1(x0 − G(x0))‖ ≤ η for some η > 0;

(C9) ‖G′x − G′y‖ ≤ ω(‖x− y‖) for all x, y ∈ D;

(C10) ‖I − L(I − G′x)‖ ≤ k for all x ∈ D and for some k ∈ (0, 13 ].

Then the sequence {xn} generated by Equation (26) is well defined, remains in Br[x0] and converges to thefixed point x∗ ∈ Br[x0] of the operator G and the sequence {xn} has R-order of convergence at least 2p + 1.

We give an example to illustrate Theorem 2.

Example 1. Let X = Y = R and D = (−1, 1) ⊂ X. Define a mapping G : D → R by

G(x) =1.1x3 − x

6(27)

for all x ∈ D. Define L : R→ R by L(x) = 7.97 x for all x ∈ R. One can easily observe that

(I − L(I − G))(x) ∈ D

for all x ∈ D. Clearly, G is differentiable on D and its derivative at x ∈ D is G′x = 3.3x2−16 and G′x is bounded

with ‖G′x‖ ≤ 0.3833 for all x ∈ D and G′ satisfies the Lipschitz condition

‖G′x − G′y‖ ≤ K‖x− y‖

for all x, y ∈ D, where K = 1.1. For x0 = 0.3, we have

z0 = (I − L(I − G))(x0) = −0.0894135714, ‖(I − G′z0)−1‖ ≤ 0.860385626 = β,

‖(I − G′z0)−1(x0 − G(x0))‖ ≤ 0.29687606 = η.

For p = 1, q = 56 and ω(t) = Kt for all t ≥ 0, we have

b0 = βω(η) = 0.280970684 < 1,

θ =

(qpbp

0p+1 + kp

)qb2

0

1− k(1 + qb0)b0= 0.0335033167 < 1

and r = 0.563139828. Hence all the conditions of Theorem 2 are satisfied. Therefore, the sequence {xn} generatedby Equation (26) is in Br[x0] and it converges to the fixed point x∗ = 0 ∈ Br[x0] of G.

200


Remark 1. In Example 1, ‖G′x‖ ≤ 0.38333 > 13 . Thus the condition (Ω) does not hold and so we can not

apply Parhi and Gupta [21] (Theorem 1) for finding fixed points of the operators like G defined by (27). Thus theStirling-like method defined by Equation (26) provides an affirmative answer of the Problem 1.

If the condition (Ω) holds, then Theorem 2 with L = I reduces to the main result of Parhi andGupta [21] as follows:

Corollary 1. [21] (Theorem 1) Let D be a nonempty open convex subset of a Banach space X and G : D → Dbe a Fréchet differentiable operator and let x0 ∈ D with z0 = G(x0). Let (I − G′z0

)−1 ∈ B(X) exists andω ∈ Φ. Assume that the conditions (C5)–(C9) and the following condition holds:

(C11) ‖G′x‖ ≤ k for all x ∈ D and for some k ∈ (0, 13 ].

Then the sequence {xn} generated by Equation (8) is well defined, remains in Br[x0] and converges to thefixed point x∗ ∈ Br[x0] of the operator G with R−order of the convergence at least 2p + 1.

Example 2. Let X = Y = R and D = (−6, 6) ⊂ X. Define a mapping G : D → R by

G(x) = 2 + esin x

5

for all x ∈ D. It is obvious that G is Fréchet differentiable on D and its Fréchet derivative at x ∈ D isG′x = cos x

5 esin x

5 . Clearly, G′x is bounded with ‖G′x‖ ≤ 0.22 < 13 = k and

‖G′x − G′y‖ ≤ K‖x− y‖

for all x, y ∈ D, where K = 0.245. For x0 = 0, we have

z0 = G(x0) = 3, ‖(I − G′z0)−1‖ ≤ 0.834725586524139 = β

and‖(I − G′z0

)−1(x0 − G(x0))‖ ≤ 2.504176759572418 = η.

For p = 1, q = 56 and ω(t) = Kt for all t ≥ 0, we have

b0 = βKη = 0.512123601526580 < 1,

θ =

(qpbp

0p+1 + kp

)qb2

0

1− k(1 + qb0)b0= 0.073280601270728 < 1

and r = 5.147038576039456.Hence all the conditions of Theorem 2 with L = I are satisfied. Therefore, the sequence {xn} generated by

Equation (26) is in Br[x0] and it converges to the fixed point x∗ = 3.023785446275295 ∈ Br[x0] of G (Table 1).

Table 1. A priori error bounds.

n ‖xn − x∗‖0 3.02378544627521 1.7795738211156 × 10−2

2 6.216484249588206 × 10−6

3 2.335501569916687 × 10−9

4 8.775202786637237 × 10−13

5 4.440892098500626 × 10−16

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4.2. Fredholm Integral Equations

Let X be a Banach space over the field F (R or C) with the norm ‖ · ‖ and D be an open convexsubset of X. Further, let B(X) be the Banach space of bounded linear operators from X into itself.Let S ∈ B(X), u ∈ X and λ ∈ F. We investigate a solution x ∈ X of the nonlinear Fredholm-typeoperator equation:

x− λSQ(x) = u, (28)

where Q : D → X is continuously Fréchet differentiable on D. The operator Equation (28) has beendiscussed in [10,38,39]. Define an operator F : D → X by

F(x) = x− λSQ(x)− u (29)

for all x ∈ D. Then solving the operator Equation (29) is equivalent to solving the operator Equation (1).From Equation (29), we have

F′x(h) = h− λSQ′x(h) (30)

for all h ∈ X. Now, we apply Theorem 1 to solve the operator Equation (28).

Theorem 3. Let X be a Banach space and D an open convex subset of X. Let Q : D → X be a continuouslyFréchet differentiable mapping at each point of D. Let L, S ∈ B(X) and u ∈ X. Assume that, for any x0 ∈ D,z0 = x0 − L(x0 − λSQ(x0)− u) and (I − λSQ′z0

)−1 exist. Assume that the condition (C6) and the followingconditions hold:

(C12) (I − L(I − λSQ))(x)− u ∈ D for all x ∈ D;(C13) ‖(I − λSQ′z0

)−1‖ ≤ β for some β > 0;(C14) ‖(I − λSQ′z0

)−1(x0 − λSQ(x0)− u)‖ ≤ η for some η > 0;(C15) ‖Q′x −Q′y‖ ≤ ω0(‖x− y‖) for all x, y ∈ D, where ω0 ∈ Φ;(C16) ω0(st) ≤ spω0(t), s ∈ [0, 1] and t ∈ [0, ∞);(C17) ‖I − L(I − λSQ′x)‖ ≤ k, k ≤ 1

3 for all x ∈ D.

Then we have the following:(1) The sequence {xn} generated by⎧⎪⎨⎪⎩

zn = xn − L(xn − λSQ(xn)− u),yn = xn − (I − λSQ′zn)

−1(xn − λSQ(xn)− u),xn+1 = yn − (I − λSQ′zn)

−1(yn − λSQ(yn)− u)(31)

for each n ∈ N0 is well defined, remains in Br[x0] and converges to a solution x∗ of the Equation (28).(2) The R-order convergence of sequence {xn} is at least 2p + 1.

Proof. Let F : D → X be an operator defined by Equation (29). Clearly, F is Fréchet differentiableat each point of D and its Fréchet derivative at x ∈ D is given by Equation (30). Now, from (C13)and Equation (30), we have ‖F′−1

z0‖ ≤ β and so it follows that (C3) holds. From (C14), Equations (29)

and (30), we have ‖F′−1z0

(F(x0))‖ ≤ η. Hence (C4) is satisfied. For all x, y ∈ D, using (C15), we have

‖F′x − F′y‖ = sup{‖(F′x − F′y)z‖ : z ∈ X, ‖z‖ = 1}≤ |λ|‖S‖ sup{‖Q′x −Q′y‖‖z‖ : z ∈ X, ‖z‖ = 1}≤ |λ|‖S‖ω0(‖x− y‖)= ω(‖x− y‖),

202


where ω(t) = |λ|‖S‖ω0(t). Clearly, ω ∈ Φ and, from (C16), we have

ω(st) ≤ spω(t)

for all s ∈ [0, 1] and t ∈ (0, ∞]. Thus (C1) and (C5) hold. (C2) follows from (C17) for c = k ∈ (0, 13 ].

Hence all the conditions of Theorem 1 are satisfied. Therefore, Theorem 3 follows from Theorem 1.This completes the proof.

Let D = X = Y = C[a, b] be the space of all continuous real valued functions defined on [a, b] ⊂ R

with the norm ‖x‖ = supt∈[a,b]

|x(t)|. Consider, the following nonlinear integral equation:

x(s) = g(s) + λ∫ b

aK(s, t)(μ(x(t))1+p + ν(x(t))2)dt (32)

for all s ∈ [a, b] and p ∈ (0, 1], where g, x ∈ C[a, b] with g(s) ≥ 0 for all s ∈ [a, b], K : [a, b]× [a, b]→ R isa continuous nonnegative real-valued function and μ, ν, λ ∈ R. Define two mappings S, Q : D → X by

Sx(s) =∫ b

aK(s, t)x(t)dt (33)

for all s ∈ [a, b] andQx(s) = μ(x(s))1+p + ν(x(s))2 (34)

for all μ, ν ∈ R and s ∈ [a, b].One can easily observe that K is bounded on [a, b]× [a, b], that is, there exists a number M ≥ 0

such that |K(s, t)| ≤ M for all s, t ∈ [a, b]. Clearly, S is bounded linear operator with ‖S‖ ≤ M(b− a)and Q is Fréchet differentiable and its Fréchet derivative at x ∈ D is given by

Q′xh(s) = (μ(1 + p)xp + 2νx)h(s) (35)

for all h ∈ C[a, b]. For all x, y ∈ D, we have

‖Q′x −Q′y‖ = sup{‖(Q′x −Q′y)h‖ : h ∈ C[a, b], ‖h‖ = 1}≤ sup{‖(μ(1 + p)(xp − yp) + 2ν(x− y))h‖ : h ∈ C[a, b], ‖h‖ = 1}≤ sup{(|μ|(1 + p)‖xp − yp‖+ 2|ν|‖x− y‖)‖h‖ : h ∈ C[a, b], ‖h‖ = 1}≤ |μ|(1 + p)‖x− y‖p + 2|ν|‖x− y‖= ω0(‖x− y‖),

(36)

where ω0(t) = |μ|(1 + p)tp + 2|ν|t, t ≥ 0 with

ω0(st) ≤ spω0(t) (37)

for all s ∈ [0, 1] and t ∈ [0, ∞). For any x ∈ D, using Equations (33) and (35), we have

‖SQ′x‖= sup{‖SQ′xh‖ : h ∈ X, ‖h‖ = 1}= sup

{sups∈[a,b]

∣∣∣∫ ba K(s, t)(μ(1 + p)(x(t))p + 2νx(t))h(t)dt

∣∣∣ : h ∈ X, ‖h‖ = 1}

≤ sup{∫ b

a |K(s, t)|(|μ|(1 + p)|x(t)|p + 2|ν||x(t)|)|h(t)|dt : h ∈ X, ‖h‖ = 1}

≤ (|μ|(1 + p)‖x‖p + 2|ν|‖x‖)M(b− a) < 1.

(38)

We now apply Theorem 3 to solve the Fredholm integral Equation (32).

203


Theorem 4. Let D = X = Y = C[a, b] and μ, ν, λ, M ∈ R. Let S, Q : D → X be operators defined byEquations (33) and (34), respectively. Let L ∈ B(X) and x0 ∈ D be such that z0 = x0 − L(x0 − λSQ(x0)−g) ∈ D. Assume that the condition (C6) and the following conditions hold:

(C18) 11−|λ|(|μ|(1+p)‖z0‖p+2|ν|‖z0‖)M(b−a) = β for some β > 0;

(C19) ‖x0−g‖+|λ|(|μ|‖x0‖p+1+2|ν|‖x0‖2)M(b−a)1−|λ|(|μ|(1+p)‖z0‖p+2|ν|‖z0‖)M(b−a) = η for some η > 0;

(C20) ‖I − L‖+ |λ|‖L‖(|μ|(1 + p)‖x‖p + 2|ν|‖x‖)M(b− a) ≤ 13

for all x ∈ D.

Then the sequence generated by Equation (31) with u = g ∈ X is well defined, remains in Br[x0] andconverges to the solution x∗ ∈ Br[x0] of the Equation (32) with R-order convergence at least (2p + 1).

Proof. Note that D = X = Y = C[a, b]. Obviously, (C12) holds. Using Equations (C20), (33),(35) and (38), we have

‖I − (I − λSQ′z0)‖ ≤ |λ|(|μ|(1 + p)‖z0‖p + 2|ν|‖z0‖)M(b− a) < 1.

Therefore, by Lemma 1, (I − λSQ′z0)−1 exists and

‖(I − λSQ′z0)−1‖ ≤ 1

1− |λ|(|μ|(1 + p)‖z0‖p + 2|ν|‖z0‖)M(b− a). (39)

Hence Equations (C18) and (39) implies (C13) holds. Using Equations (C19), (38) and (39),we have

‖(I − λSQ′z0)−1(x0 − λSQ(x0)− g)‖

≤ ‖(I − λSQ′z0)−1‖(‖x0 − g‖+ ‖λSQ(x0)‖)

≤ ‖x0 − g‖+ |λ|(|μ|‖x0‖p+1 + 2|ν|‖x0‖2)M(b− a)1− |λ|(|μ|(1 + p)‖z0‖p + 2|ν|‖z0‖)M(b− a)

≤ η.

Thus the condition (C14) is satisfied. The conditions (C15) and (C16) follow from Equations (36)and (37), respectively. Now, from Equation (C20) and (38), we have

‖I − L(I − λSQ′x)‖ ≤ ‖I − L‖+ ‖L‖‖λSQ′x‖≤ ‖I − L‖+ ‖L‖|λ|(|μ|(1 + p)‖x‖p + 2|ν|‖x‖)M(b− a)

≤ 13

.

This implies that (C17) holds. Hence all the conditions of Theorem 3 are satisfied. Therefore,Theorem 4 follows from Theorem 3. This completes the proof.

Now, we give one example to illustrate Theorem 3.

Example 3. Let X = Y = C[0, 1] be the space of all continuous real valued functions defined on [0, 1].Let D = {x : x ∈ C[0, 1], ‖x‖ < 3

2} ⊂ C[0, 1]. Consider the following nonlinear integral equation:

x(s) = sin(πs) +110

∫ 1

0cos(πs) sin(πt)(x(t))p+1dt, p ∈ (0, 1]. (40)

Define two mappings S : X → X and Q : D → Y by

S(x)(s) =∫ 1

0K(s, t)x(t)dt, Q(x)(s) = (x(s))p+1,

204


where K(s, t) = cos(πs) sin(πt). For u = sin(πs), the problem Equation (40) is equivalent to the problemEquation (28). Here, one can easily observe that S is bounded linear operator with ‖S‖ ≤ 1 and Q is Fréchetdifferentiable with Q′xh(s) = (p + 1)(x(s))ph(s) for all h ∈ X and s ∈ [0, 1]. For all x, y ∈ D, we have

‖Q′x −Q′y‖ ≤ (p + 1)‖x− y‖p = ω0(‖x− y‖),

where ω0(t) = (p + 1)tp for any t ≥ 0. Clearly, ω0 ∈ Φ. Define a mapping F : D → X by

F(x)(s) = x(s)− 110

SQ(x)(s)− sin(πs).

Clearly, F is Fréchet differentiable on D. We now show that (C12) holds for L = I ∈ B(X). Note that

‖(I − L(I − λSQ))(x)− u‖ =∥∥∥∥ 1

10SQ(x)(s) + sin(πs)

∥∥∥∥ ≤ 110

(32

)p+1+ 1 <

32

for all x ∈ D. Thus (I − L(I − λSQ))(x)− u ∈ D for all x ∈ D. For all x ∈ D, we have

‖I − F′x‖ ≤p + 1

10‖x‖p ≤ p + 1

10

(32

)p= k.

Therefore, by Lemma 1, F′−1x exists and

F′−1x U(s) = U(s) +

(p + 1) cos(πs)∫ 1

0 sin(πt)(x(t))pU(t)dt

10− (p + 1)∫ 1

0 sin(πt) cos(πt)(x(t))pdt(41)

for all U ∈ Y.

Let x0(s) = sin(πs), ω(t) = ω0(t)10 = p+1

10 tp . Then we have the following:

(a) x0 ∈ X, F(x0(s)) = − cos(πs)10

∫ 10 (sin(πt))p+2dt;

(b) z0(s) = x0(s)− F(x0(s)) = sin(πs) + cos(πs)10

∫ 10 (sin(πt))p+2dt;

(c) ‖F′−1z0‖ ≤ 10p+1

10p+1−(p+1)11p = β;

(d) ‖F′−1z0

F(x0)‖ ≤ 10p

10p+1−(p+1)11p = η;

(e) b0 = βω(η) = (p+1)10p(p+1)

(10p+1−(p+1)11p)p+1 and q = 1p+1 +

(p+110

)p ( 32)p2

.

One can easily observe that θ =

(qb0

2 +k)

qb02

1−k(1+q)b0< 1 for all p ∈ (0, 1] and r = (1+qb0)η

1−θ . Hence all theconditions of Theorem 3 are satisfied. Therefore, the sequence {xn} generated by Equation (31) iswell defined, remains in Br[x0] and converges to a solution of the integral Equation (40).

For p = 1, the convergence behavior of Newton-like method Equation (31) is given in Table 2.

Table 2. Iterates of Newton-like method Equation (31).

n xn(s) zn(s) yn(s)

0 sin(πs) sin(πs) + 0.0424413182 cos(πs) sin(πs) + 0.0425947671 cos(πs)1 sin(πs) + 0.0424794035 cos(πs) sin(πs) + 0.0424791962 cos(πs) sin(πs) + 0.0424796116 cos(πs)2 sin(πs) + 0.0424795616 cos(πs) sin(πs) + 0.042479611 cos(πs) sin(πs) + 0.0424796112 cos(πs)3 sin(πs) + 0.0424796111 cos(πs) sin(πs) + 0.0424796109 cos(πs) sin(πs) + 0.0424796113 cos(πs)

5. Conclusions

The semilocal convergence of the third order Newton-like method for finding zeros of an operatorfrom a Banach space to another Banach space and the corresponding Stirling-like method for finding

205


fixed points of an operator on a Banach space are established under the ω-continuity condition.Our iterative technique is applied to nonlinear Fredholm-type operator equations. The R-order of ourmethods are clearly shown to be equal to at least 2p + 1 for any p ∈ (0, 1]. Some numerical examplesare given in support of our work, where earlier work cannot apply. In future, our iterative techniquescan be applied in optimization problems.

Author Contributions: The authors have contributed equally to this paper.

Funding: This research has received no exteranl funding.

Acknowledgments: We would like to thank the reviewers for their valuable suggestions for improvement ofthis paper.


Reference

1. Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Aregamon Press: Oxford, UK, 1982.2. Kantorovich, L.V. On Newton’s method for functional equations. Dokl. Akad. Nauk. SSSR 1948, 59, 1237–1240.

(In Russian)3. Rheinbolt, W.C. A unified Convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 1968,

5, 42–63. [CrossRef]4. Argyros, I.K.; Cho, Y.J.; Hilout, S. Numerical Methods for Equations and its Applications; CRC Press/Taylor &

Francis Group Publ. Comp.: New York, NY, USA, 2012.5. Argyros, I.K.; Hilout, S. Improved generaliged differentiability conditions for Newton-like methods.

J. Complex. 2010, 26, 316–333. [CrossRef]6. Argyros, I.K.; Hilout, S. Majorizing sequences for iterative methods. J. Comput. Appl. Math. 2012, 236,

1947–1960. [CrossRef]7. Argyros, I.K. An improved error analysis for Newton-like methods under generalized conditions. J. Comput.

Appl. Math. 2003, 157, 169–185. [CrossRef]8. Argyros, I.K.; Hilout, S. On the convergence of Newton-type methods under mild differentiability conditions.

Number. Algorithms 2009, 52, 701–726. [CrossRef]9. Sahu, D.R.; Singh, K.K.; Singh, V.K. Some Newton-like methods with sharper error estimates for solving

operator equations in Banach spaces. Fixed Point Theory Appl. 2012, 78, 1–20. [CrossRef]10. Sahu, D.R.; Singh, K.K.; Singh, V.K. A Newton-like method for generalized operator equations in Banach

spaces. Numer. Algorithms 2014, 67, 289–303. [CrossRef]11. Sahu, D.R.; Cho, Y.J.; Agarwal, R.P.; Argyros, I.K. Accessibility of solutions of operator equations by

Newton-like Methods. J. Comlex. 2015, 31, 637–657. [CrossRef]12. Argyros, I.K. A unifying local-semilocal convergence analysis and applications for two-point Newton-like

methods in Banach space. J. Math. Anal. Appl. 2004, 298, 374–397. [CrossRef]13. Argyros, I.K.; Cho, Y.J.; George, S. On the “Terra incognita” for the Newton-Kantrovich method. J. Korean

Math. Soc. 2014, 51, 251–266. [CrossRef]14. Argyros, I.K.; Cho, Y.J.; George, S. Local convergence for some third-order iterative methods under weak

conditions. J. Korean Math. Soc. 2016, 53, 781–793. [CrossRef]15. Ren, H.; Argyros, I.K.; Cho, Y.J. Semi-local convergence of Steffensen-type algorithms for solving nonlinear

equations. Numer. Funct. Anal. Optim. 2014, 35, 1476–1499. [CrossRef]16. Ezquerro, J.A.; Hernández, M.A.; Salanova, M.A. A discretization scheme for some conservative problems.

J. Comput. Appl. Math. 2000, 115, 181–192. [CrossRef]17. Ezquerro, J.A.; Hernández, M.A.; Salanova, M.A. A newton like method for solving some boundery value

problems. Numer. Funct. Anal. Optim. 2002, 23, 791–805. [CrossRef]18. Ezquerro, J.A.; Hernández, M.A. Generalized differentiability conditions for Newton’s method. IMA J.

Numer. Anal. 2002, 22, 187–205. [CrossRef]19. Proinov, P.D. New general convergence theory for iterative process and its applications to Newton-

Kantorovich type theores. J. Complex. 2010, 26, 3–42. [CrossRef]20. Sahu, D.R.; Yao, J.C.; Singh, V.K.; Kumar, S. Semilocal Convergence Analysis of S-iteration Process of

Newton-Kantorovich Like in Banach Spaces. J. Optim. Theory Appl. 2017, 172, 102–127. [CrossRef]

206


21. Parhi, S.K.; Gupta, D.K. Convergence of a third order method for fixed points in Banach spaces.Numer. Algorithms 2012, 60, 419–434. [CrossRef]

22. Agarwal, R.P.; O’Regan, D.; Sahu, D.R. Fixed Point Theory for Lipschitzian-Type Mappings with Applications;Topological Fixed Point Theory and its Applications; Springer: New York, NY, USA, 2009; p. 6.

23. Agarwal, R.P.; Meehan, M.; O’Regan, D. Fixed Point Theory and Applications; Cambridge University Press:Cambridge, UK, 2004.

24. Picard, E. Memorire sur la theorie des equations aux derivees partielles et la methode aes approximationssuccessive. J. Math. Pures Appl. 1980, 6, 145–210.

25. Cho, Y.J. Survey on metric fixed point theory and applications. In Advances on Real and Complex Analysiswith Applications; Trends in Mathematics; Ruzhahsky, M., Cho, Y.J., Agarwal, P., Area, I., Eds.; Birkhäuser,Springer: Basel, Switzerland, 2017.

26. Granas, A.; Dugundji, J. Fixed Point Theory; Springer: New York, NY, USA, 2003.27. Zeidler, E. Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems; Springer:

New York, NY, USA, 1986.28. Zeidler, E. Nonlinear Functional Analysis and Its Applications III: Variational Methods and Applications; Springer:

New York, NY, USA, 1985.29. Argyros, I.K. On Newton’s method under mild differentiability conditions and applications. Appl. Math. Comput.

1999, 102, 177–183. [CrossRef]30. Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press:

New York, NY, USA; London, UK, 1970.31. Bartle, R.G. Newton’s method in Banach spaces. Proc. Am. Math. Soc. 1955, 6, 827–831.32. Rall, L.B. Convergence of Stirling’s method in Banaeh spaces. Aequa. Math. 1975, 12, 12–20. [CrossRef]33. Parhi, S.K.; Gupta, D.K. Semilocal convergence of a Stirling-like method in Banach spaces. Int. J. Comput. Methods

2010, 7, 215–228. [CrossRef]34. Argyros, I.K.; Muruster, S.; George, S. On the Convergence of Stirling’s Method for Fixed Points Under Not

Necessarily Contractive Hypotheses. Int. J. Appl. Comput. Math. 2017, 3, 1071–1081. [CrossRef]35. Alshomrani, A.S.; Maroju, P.; Behl, R. Convergence of a Stirling-like method for fixed points in Banach

spaces. J. Comput. Appl. Math. 2018. [CrossRef]36. Parhi, S.K.; Gupta, D.K. A third order method for fixed points in Banach spaces. J. Math. Anal. Appl. 2009,

359, 642–652. [CrossRef]37. Rall, L.B. Computational Solution of Nonlinear Operator Equations; John Wiley and Sons: New York, NY, USA, 1969.38. Hernández, M.A.; Salanova, M.A. A Newton-like iterative process for the numerical solution of Fredholm

nonlinear integral equations. J. Integr. Equ. Appl. 2005, 17, 1–17. [CrossRef]39. Kohaupt, L. A Newton-like method for the numerical solution of nonlinear Fredholm-type operator

equations. Appl. Math. Comput. 2012, 218, 10129–10148. [CrossRef]


207

mathematics

Article

Ball Comparison for Some Efficient Fourth OrderIterative Methods Under Weak Conditions

Ioannis K. Argyros 1,† and Ramandeep Behl 2,*,†

1 Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA; [email protected] Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia* Correspondence: [email protected]; Tel.: +96-570-811-650† These authors contributed equally to this work.

Received: 17 December 2018; Accepted: 9 January 2019; Published: 16 January 2019

Abstract: We provide a ball comparison between some 4-order methods to solve nonlinear equationsinvolving Banach space valued operators. We only use hypotheses on the first derivative, as comparedto the earlier works where they considered conditions reaching up to 5-order derivative, althoughthese derivatives do not appear in the methods. Hence, we expand the applicability of them.Numerical experiments are used to compare the radii of convergence of these methods.

Keywords: fourth order iterative methods; local convergence; banach space; radius of convergence

MSC: 65G99; 65H10; 47H17; 49M15

1. Introduction

Let E1, E2 be Banach spaces and D ⊂ E1 be a nonempty and open set. Set LB(E1, E2) = {M :E1 → E2}, bounded and linear operators. A plethora of works from numerous disciplines can bephrased in the following way:

λ(x) = 0, (1)

using mathematical modelling, where λ : D → E2 is a continuously differentiable operator in theFréchet sense. Introducing better iterative methods for approximating a solution s∗ of expression (1)is a very challenging and difficult task in general. Notice that this task is extremely important, sinceexact solutions of Equation (1) are available in some occasions.

We are motivated by four iterative methods given as⎧⎪⎨⎪⎩yj =xj − 2

3λ′(xj)

−1λ(xj)

xn+1 =xj − 12

[(3λ′(yj)− λ′(xj)

)−1(3λ′(yj) + λ′(xj)

)]λ′(xj)

−1λ(xj),(2)

⎧⎪⎨⎪⎩yj =xj − 2

3λ′(xj)

−1λ(xj)

xn+1 =xj −[− 1

2I +

98

Bj +38

Aj

]λ′(xj)

−1λ(xj),(3)

⎧⎪⎨⎪⎩yj =xj − 2

3λ′(xj)

−1λ(xj)

xn+1 =xj −[

I +14(Aj − I) +

38(Aj − I)2

]λ′(yj)

−1λ(xj),(4)



and ⎧⎨⎩ yj =xj − Hjλ′(xj)

−1λ(xj)

xn+1 =zj −[3I − Hjλ

′(xj)−1[xj, zj; λ]

]λ′(xj)

−1λ(zj),(5)

where H0j = H0(xj), x0, y0 ∈ D are initial points, H(x) = 2I + H0(x), Hj = H(Xj) ∈ LB(E1,E1),

Aj = λ′(xj)−1λ′(yj), zj =

xj+yj2 , Bj = λ′(yj)

−1λ′(xj), and [·, ·; λ] : D×D→ LB(E1,E1) is a first orderdivided difference. These methods specialize to the corresponding ones (when E1 = E2 = Ri, i isa natural number) studied by Nedzhibov [1], Hueso et al. [2], Junjua et al. [3], and Behl et al. [4],respectively. The 4-order convergence of them was established by Taylor series and conditions onthe derivatives up to order five. Even though these derivatives of higher-order do not appear in themethods (2)–(5). Hence, the usage of methods (2)–(5) is very restricted. Let us start with a simpleproblem. Set E1 = E2 = R and D = [− 5

2 , 32 ]. We suggest a function λ : A→ R as

λ(t) =

{0, t = 0t5 + t3 ln t2 − t4, t �= 0

.

Then, s∗ = 1 is a zero of the above function and we have

λ′(t) = 5t4 + 3t2 ln t2 − 4t3 + 2t2,

λ′′(t) = 20t3 + 6t ln t2 − 12t2 + 10t,

andλ′′′(t) = 60t2 + 6 ln t2 − 24t + 22.

Then, the third-order derivative of function λ′′′(x) is not bounded on D. The methods (2)–(5)cannot be applicable to such problems or their special cases that require the hypotheses on the third orhigher-order derivatives of λ. Moreover, these works do not give a radius of convergence, estimationson ‖xj − s∗‖, or knowledge about the location of s∗. The novelty of our work is that we provide thisinformation, but requiring only the derivative of order one, for these methods. This expands thescope of utilization of them and similar methods. It is vital to note that the local convergence resultsare very fruitful, since they give insight into the difficult operational task for choosing the startingpoints/guesses.

Otherwise with the earlier approaches: (i) We use the Taylor series and high order derivative,(ii) we do not have any clue for the choice of the starting point x0, (iii) we have no estimate in advanceabout the number of iterations needed to obtain a predetermined accuracy, and (iv) we have noknowledge of the uniqueness of the solution.

The work lays out as follows: We give the convergence of these iterative schemes (2)–(5) withsome main theorems in Section 2. Some numerical problems are discussed in the Section 3. The finalconclusions are summarized in Section 4.

2. Local Convergence Analysis

Let us consider that I = [0, ∞) and ϕ0 : I → I be a non-decreasing and continuous function withϕ0(0) = 0.

Assume that the following equationϕ0(t) = 1 (6)

has a minimal positive solution ρ0. Let I0 = [0, ρ0). Let ϕ : I0 → I and ϕ1 : I0 → I be continuous andnon-decreasing functions with ϕ(0) = 0. We consider functions on the interval I0 as

ψ1(t) =

∫ 10 ϕ

((1− τ)t

)dτ + 1

3

∫ 10 ϕ1(τt)dτ

1− ϕ0(t)

209


andψ1(t) = ψ1(t)− 1.

Suppose thatϕ0(t) < 3. (7)

Then, by (7), ψ1(0) < 0 and ψ1(t) → ∞, as t → ρ−0 . On the basis of the classical intermediatevalue theorem, the function ψ1(t) has a minimal solution R1 in (0, ρ0). In addition, we assume

q(t) = 1 (8)

has a minimal positive solution ρq, where

q(t) =12

(3ϕ0(ψ1(t)t) + ϕ0(t)

).

Set ρ = min{ρ0, ρq}.Moreover, we consider two functions ψ2 and ψ2 on I1 = [0, ρ) as

ψ2(t) =

∫ 10 ϕ

((1− τ)t

)dτ

1− ϕ0(t)+

32

(ϕ0(ψ1(t)t

)+ ϕ0(t)

) ∫ 10 ϕ1(τt)dτ

(1− q(t))(1− ϕ0(t))

andψ2(t) = ψ2(t)− 1.

Then, ψ2(0) = −1, and ψ2(t) → ∞, with t → ρ−. We recall R2 as the minimal solution ofψ2(t) = 0. Set

R = min{R1, R2}. (9)

It follows from (9) that for every t ∈ [0, R)

0 ≤ ϕ0(t) < 1, (10)

0 ≤ ψ1(t) < 1, (11)

0 ≤ q(t) < 1 (12)

and0 ≤ ψ2(t) < 1 (13)

Define by S(s∗, r) ={

y ∈ E1 : ‖s∗ − y‖ < r,}

and denote by S(s∗, r) the closure of S(s∗, r).The local convergence of method (2) uses the conditions (A):

(a1) λ : D→ E2 is a continuously differentiable operator in the Fréchet sense, and there exists s∗ ∈ D.(a2) There exists a function ϕ0 : I → I non-decreasing and continuous with ϕ0(0) = 0 for all x ∈ D∥∥∥λ′(s∗)−1

(λ′(x)− λ′(s∗)

)∥∥∥ ≤ ϕ0(‖x− s∗‖).

Set D0 = D∩ S(s∗, ρ0), where ρ0 is given in (6).(a3) There exist functions ϕ : I0 → I, ϕ1 : I0 → I non-decreasing and continuous with ϕ(0) = 0 so

that for all x, y ∈ D0 ∥∥∥λ′(s∗)−1(

λ′(y)− λ′(x))∥∥∥ ≤ ϕ(‖y− x‖)

and ∥∥∥λ′(s∗)−1λ′(x)∥∥∥ ≤ ϕ1(‖y− x‖)

210


(a4) S(s∗, R) ⊂ D, radii ρ0, ρq as given, respectively by (6), (8) exist; the condition (7) holds, where Ris defined in (9).

(a5) ∫ 1

0ϕ0(τR∗)dτ < 1, for some R∗ ≥ R.

Set D1 = D ∩ S(s∗, R∗).

We can now proceed with the local convergence study of Equation (2) adopting the precedingnotations and the conditions (A).

Theorem 1. Under the conditions (A) sequence {xj} starting at x0 ∈ S(s∗, R) − {s∗} converges to s∗,{xj} ⊂ S(x, R) so that

‖yj − s∗‖ ≤ ψ1(‖xj − s∗‖)‖xj − s∗‖ ≤ ‖xj − s∗‖ < R (14)

and‖xn+1 − s∗‖ ≤ ψ2(‖xj − s∗‖)‖xj − s∗‖ ≤ ‖xj − s∗‖, (15)

with ψ1 and ψ2 functions considered previously and R is given in (9). Moreover, s∗ is a unique solution in theset D1.

Proof. We proof the estimates (14) and (15) by adopting mathematical induction. Therefore, weconsider x ∈ S(s∗, R)− {s∗}. By (a1), (a2), (9), and (10), we have

‖λ′(s∗)−1(λ′(s∗)− λ′(x))‖ ≤ ϕ0(‖s∗ − x0‖) < ϕ0(R) < 1, (16)

hence λ′(x)−1 ∈ LB(E2,E1) and

‖λ′(x)−1λ′(s∗)‖ ≤ 11− ϕ0(‖s∗ − x0‖) . (17)

The point y0 is also exists by (17) for n = 0. Now, by using (a1), we have

λ(x) = λ(x)− λ(s∗) =∫ 1

0λ′(s∗ + τ(x− s∗))dτ(x− s∗). (18)

From (a3) and (18), we yield∥∥∥λ′(s∗)−1λ(x)∥∥∥ ≤ ∫ 1

0ϕ1(τ‖x− s∗‖)dτ‖x− s∗‖. (19)

We can also write by method (2) for n = 0

y0 − s∗ =(

x0 − s∗ − λ′(x0)−1λ(x0)

)+

13

λ′(x0)−1λ(x0). (20)

By expressions (9), (11), (17), (19), and (20), we obtain in turn that

‖y0 − s∗‖ ≤∥∥∥λ′(x0)

−1λ′(s∗)∥∥∥ ∥∥∥∥∫ 1

0λ′(s∗)−1

(λ′(s∗ + τ(x0 − s∗)

)− λ′(x0))(x0 − s∗)dτ

∥∥∥∥+

13

∥∥∥λ′(x0)−1λ′(s∗)

∥∥∥ ∥∥∥λ′(s∗)−1λ(x0)∥∥∥

≤∫ 1

0 ϕ((1− τ)‖x0 − s∗‖

)dτ + 1

3

∫ 10 ϕ(τ‖x0 − s∗‖)dτ

1− ϕ0(‖x0 − s∗‖)= ψ1(‖x0 − s∗‖)‖x0 − s∗‖ ≤ ‖x0 − s∗‖ < R,

(21)

211


which confirms y0 ∈ S(s∗, R) and (14) for n = 0. We need to show that(

3λ′(y0)− 3λ′(x0))−1 ∈

LB(E2,E1).In view of (a2), (12), and (21), we have∥∥∥(2λ′(s∗)

)−1[3λ′(y0)− λ′(x0)− 3λ′(s∗) + λ′(s∗)

]∥∥∥≤ 1

2

[3∥∥∥λ′(s∗)−1(λ′(y0)− λ′(s∗)

)∥∥∥+ ∥∥∥λ′(s∗)−1(λ′(x0)− λ′(s∗))∥∥∥ ]

≤ 12

[ϕ0(‖y0 − s∗‖) + ϕ0(‖x0 − s∗‖)

]≤ 1

2

[ϕ0(ψ1(‖x0 − s∗‖)‖x0 − s∗‖

)+ ϕ0(‖x0 − s∗‖)

]= q(‖x0 − s∗‖) < 1,

(22)

so ∥∥∥∥(3λ′(y0)− λ′(x0))−1

λ′(s∗)∥∥∥∥ ≤ 1

1− q(‖x0 − s∗‖) . (23)

Using (9), (13), (17), (a3), (21), (23), and the second substep of method (2) (since x1 exists by (23)),we can first write

x1 − s∗ = x0 − s∗ − λ′(x0)−1λ(x0)

+[

I − 12(3λ′(y0)− λ′(x0)

)−1(3λ′(y0) + λ′(x0))]

λ′(x0)−1λ(x0)

(24)

so

‖x1 − s∗‖ ≤ ‖x0 − s∗ − λ′(x0)−1λ(x0)‖+ 3

2‖(3λ′(y0)− λ′(x0)

)−1λ′(s∗)‖

×[‖λ′(s∗)−1(λ′(y0)− λ′(x0)

)‖+ ‖λ′(s∗)−1(λ′(x0)− λ′(s∗))−1‖

]‖λ′(x0)

−1λ(s∗)‖‖λ′(x0)−1λ(x0)‖

≤

⎡⎢⎣ ∫ 10 ϕ((1− τ)t)dτ

1− ϕ0(t)+

32

(ϕ0(‖y0 − s∗‖) + ϕ0(‖x0 − s∗‖)

) ∫ 10 ϕ1(τ‖x0 − s∗‖)dτ

(1− q(‖x0 − s∗‖))(1− ϕ0(‖x0 − s∗‖))

⎤⎥⎦ ‖x0 − s∗‖

≤ ψ2(‖x0 − s∗‖)‖x0 − s∗‖ ≤ ‖x0 − s∗‖.

(25)

So, (15) holds and x1 ∈ S(s∗, R).To obtain estimate (25), we also used the estimate

I − 12(3λ′(y0)− λ′(x0)

)−1(3λ′(y0) + λ′(x0))

=12(3λ′(y0)− λ′(x0)

)−1[2(3λ′(y0)− λ′(x0)

)− (3λ′(y0) + λ′(x0))]

=32(3λ′(y0)− λ′(x0)

)−1[(

λ′(y0)− λ′(s∗))+(λ′(s∗)− λ′(x0)

)] (26)

The induction for (14) and (15) can be finished, if xm, ym, xm+1 replace x0, y0, x1 in the precedingestimations. Then, from the estimate

‖xm+1 − s∗‖ ≤ μ‖xm − s∗‖ < R, μ = ϕ2(‖x0 − s∗‖) ∈ [0, 1), (27)

we arrive at limm→∞

xm = s∗ and xm+1 ∈ S(s∗, R). Let us consider that K =∫ 1

0 λ′(y∗ + τ(s∗ − y∗))dτ for

y∗ ∈ D1 with K(y∗) = 0. From (a1) and (a5), we obtain

‖λ′(s∗)−1(λ′(s∗)− K)‖ ≤ ∫ 10 ϕ0(τ‖s∗ − y∗‖)dτ

≤ ∫ 10 ϕ0(τR)dτ < 1.

(28)

212


So, K−1 ∈ LB(E1,E2), and s∗ = y∗ by the identity

0 = K(s∗)− K(y∗) = K(s∗ − y∗). (29)

Proof. Next, we deal with method (3) in an analogous way. We shall use the same notation aspreviously. Let ϕ0, ϕ, ϕ1, ρ0, ψ1, R1, and ψ1, be as previously.

We assumeϕ0(ψ1(t)t

)= 1 (30)

has a minimal solution ρ1. Set ρ = min{ρ0, ρ1}. Define functions ψ2 and ψ2 on interval I2 = [0, ρ) by

ψ2(t) =

∫ 10 ϕ

((1− τ)t

)dτ

1− ϕ0(t)+

⎡⎢⎣2 +3(

ϕ0(ψ1(t)t) + ϕ0(t))

8(1− ϕ0(t))+

9(

ϕ0(ψ1(t)t) + ϕ0(t))

8(1− ϕ0(ψ1(t)t))

⎤⎥⎦ ∫ 10 ϕ1(τt)dτ

1− ϕ0(t)

andψ2(t) = ψ2(t)− 1.

Then, ψ2(0) = −1 and ψ2(t)→ ∞, with t → ρ−. R2 is known as the minimal solution of equationψ2(t) = 0 in (0, ρ), and set

R = min{R1, R2}. (31)

Replace ρq by ρ1 in the conditions (A) and call the resulting conditions (A)′.Moreover, we use the estimate obtained for the second substep of method (3)

x1 − s∗ = x0 − s∗ − λ′(x0)−1λ(x0) +

[32

I − 98

B0 − 916

A0

]λ′(x0)

−1λ(x0)

= x0 − s∗ − λ′(x0)−1λ(x0) +

[− 2I +

38(I − A0) +

98(I − B0)

]λ′(x0)

−1λ(x0)

= x0 − s∗ − λ′(x0)−1λ(x0) +

[− 2I +

38

λ′(x0)−1(

λ′(x0)− λ′(y0))

+98

λ′(y0)−1(

λ′(y0)− λ′(x0))]

λ′(x0)−1λ(x0).

(32)

Then, by replacing (24) by (32) in the proof of Theorem 1, we have instead of (25)

‖x1 − s∗‖ =[∫ 1

0 ϕ((1− τ)‖s∗ − x0‖)dτ

1− ϕ(‖s∗ − x0‖) +

{2 +

3(

ϕ(‖y0 − s∗‖) + ϕ0(‖s∗ − x0‖))

8(1− ϕ0(‖s∗ − x0‖))

+9(

ϕ0(‖y0 − s∗‖) + ϕ0(‖s∗ − x0‖))

8(1− ϕ0(‖y0 − s∗‖))

}∫ 10 ϕ1(‖s∗ − x0‖)dτ

1− ϕ0(‖s∗ − x0‖)

]‖s∗ − x0‖

≤ ψ2(‖s∗ − x0‖)‖s∗ − x0‖ ≤ ‖s∗ − x0‖.

(33)

The rest follows as in Theorem 1.

Hence, we arrived at the next Theorem.

Theorem 2. Under the conditions (A)′, the conclusions of Theorem 1 hold for method (3).

213


Proof. Next, we deal with method (4) in the similar way. Let ϕ0, ϕ, ϕ1, ρ0, ρ1, ρ, ψ1, R1, and ψ1, be as inthe case of method (3). We consider functions ψ2 and ψ2 on I1 as

ψ2(t) =

∫ 10 ϕ((1− τ)t)dτ

1− ϕ0(t)+

ϕ0(ψ1(t)t) + ϕ0(t)(1− ϕ0(t)

)(1− ϕ0(ψ1(t)t)

) + 14

(ϕ0(ψ1(t)t) + ϕ0(t)

)(1− ϕ0(t)

)+

38

(ϕ0(ψ1(t)t) + ϕ0(t)

1− ϕ0(t)

)2

andψ2(t) = ψ2(t)− 1.

The minimal zero of ψ2(t) = 0 is denoted by R2 in (0, ρ), and set

R = min{R1, R2}. (34)

Notice again that from the second substep of method (4), we have

x1 − s∗ = x0 − s∗ − λ′(x0)−1λ(x0) +

[λ′(x0)

−1 − λ′(y0)−1 − 1

4(A0 − I)− 3

8(I − A0)

2]λ(x0)

= x0 − s∗ − λ′(x0)−1λ(x0) +

{λ′(x0)

−1[(

λ′(y0)− λ′(s∗))+(λ′(s∗)− λ′(x0)

)]− 1

4λ′(x0)

−1[(

λ′(y0)− λ′(s∗))+(λ′(s∗)− λ′(x0)

)]− 3

8λ′(x0)

−1[(

λ′(y0)− λ′(s∗))+(λ′(s∗)− λ′(x0)

)]2}

λ(x0),

(35)

so

‖x1 − s∗‖ ≤[∫ 1

0 ϕ((1− τ)‖s∗ − x0‖)dτ

1− ϕ(‖s∗ − x0‖) +ϕ0(ψ(‖s∗ − x0‖)‖s∗ − x0‖

)+ ϕ0(‖s∗ − x0‖)

(1− ϕ0(‖s∗ − x0‖))(1− ϕ0(ψ(‖s∗ − x0‖)‖s∗ − x0‖)

)+

14

(ϕ(ψ1(‖s∗ − x0‖)‖s∗ − x0‖

)+ ϕ0(‖s∗ − x0‖)

)(1− ϕ0(‖s∗ − x0‖)

)+

38

(ϕ(ψ1(‖s∗ − x0‖)‖s∗ − x0‖

)+ ϕ0(‖s∗ − x0‖)(

1− ϕ0(‖s∗ − x0‖)) )2 ]

‖s∗ − x0‖

≤ ψ2(‖s∗ − x0‖)‖s∗ − x0‖ ≤ ‖s∗ − x0‖.

(36)

The rest follows as in Theorem 1.

Hence, we arrived at the next following Theorem.

Theorem 3. Under the conditions (A)′, conclusions of Theorem 1 hold for scheme (4).

Proof. Finally, we deal with method (5). Let ϕ0, ϕ, ϕ1, ρ0, I0 be as in method (2). Let also ϕ2 : I0 → I,ϕ3 : I0 → I, ϕ4 : I0 → I and ϕ5 : I0 × I0 → I be continuous and increasing functions with ϕ3(0) = 0.We consider functions ψ1 and ψ1 on I0 as

ψ1(t) =

∫ 10 ϕ((1− τ)t)dτ + ϕ2(t)

∫ 10 ϕ1(τt)dτ

1− ϕ0(t)

andψ1(t) = ψ1(t)− 1.

214


Suppose thatϕ1(0)ϕ2(0) < 1. (37)

Then, by (6) and (37), we yield ψ1(0) < 0 and ψ1(t) → ∞ with t → ρ−0 . R1 is known as theminimal zero of ψ1(t) = 0 in (0, ρ0). We assume

ϕ0(

g(t)t)= 1, (38)

where g(t) = 12(1 + ψ1(t)

), has a minimal positive solution ρ1. Set I1 = [0, ρ), where ρ = min{ρ0, ρ1}.

We suggest functions ψ2 and ψ2 on I1 as

ψ2(t) =

[∫ 10 ϕ((1− τ)g(t)t)dτ

1− ϕ0(g(t)t)+

(ϕ0(g(t)t) + ϕ0(t)

) ∫ 10 ϕ1(τg(t)t)dτ

(1− ϕ0(g(t)))(1− ϕ0(t))

+ 2ϕ3( t

2 (1 + ψ1(t))) ∫ 1

0 ϕ1(τg(t)t)dτ

(1− ϕ0(t))2 +ϕ4(t)ϕ5(t, ψ1(t)t)

∫ 10 ϕ1(τg(t)t)dτ

(1− ϕ0(t))2

]g(t)

andψ2(t) = ψ2(t)− 1.

Suppose that (2ϕ3(0) + ϕ4(0)ϕ5(0, 0)

)ϕ1(0) < 1. (39)

By (39) and the definition of I1, we have ψ2(0) < 0, ψ2(t) → ∞ with t → ρ−. We assume R2 asthe minimal solution of ψ2(t) = 0. Set

R = min{R1, R2}. (40)

The study of local convergence of scheme (5) is depend on the conditions (C):

(c1) = (a1).(c2) = (a2).(c3) There exist functions ϕ : I1 → I, ϕ1 : I0 → I, ϕ2 : I0 → I, ϕ3 : I0 → I, ϕ4 : I0 → I, and

ϕ5 : I0 × I0 → I, increasing and continuous functions with ϕ(0) = ϕ3(0) = 0 so for all x, y ∈ D0

‖λ′(s∗)−1(λ′(y)− λ′(x))‖ ≤ ϕ(‖y− x‖),

‖λ′(s∗)−1λ′(x)‖ ≤ ϕ1(‖x− s∗‖),‖I − H(x)‖ ≤ ϕ2(‖x− s∗‖),‖λ′(s∗)−1([x, y; λ]− λ′(x)

)‖ ≤ ϕ3(‖y− x‖),‖H0(x)‖ ≤ ϕ4(‖x− s∗‖),

and‖λ′(s∗)−1[x, y; λ]‖ ≤ ϕ5(‖x− s∗‖, ‖y− s∗‖),

(c4) S(s∗, R) ⊆ D, ρ0, ρ1 given, respectively by (6), (38) exist, (37) and (38) hold, and R is definedin (40).

(c5) = (a5).

215


Then, using the estimates

‖y0 − s∗‖ = ‖x0 − s∗ − λ′(x0)−1λ(x0) + (I − H0)λ

′(x0)−1λ(x0)‖

≤∫ 1

0 ϕ((1− τ)‖x0 − s∗‖)dτ‖x0 − s∗‖1− ϕ0(‖x0 − s∗‖) + ‖I − H0‖‖λ′(x0)

−1λ′(s∗)‖‖λ′(s∗)−1λ(x0)‖

≤[∫ 1

0 ϕ((1− τ)‖x0 − s∗‖)dτ + ϕ2(‖x0 − s∗‖)∫ 1

0 ϕ1(τ‖x0 − s∗‖)dτ

1− ϕ0(‖x0 − s∗‖)

]‖x0 − s∗‖

≤ ψ1(‖x0 − s∗‖)‖x0 − s∗‖ ≤ ‖x0 − s∗‖ < R,

(41)

and

‖x1 − s∗‖ = ‖z0 − s∗ − λ′(z0)−1λ(z0) + λ′(z0)

−1(λ′(x0)− λ′(z0))λ′(x0)

−1λ(z0)‖+ 2λ′(x0)

−1([x0, z0; λ]− λ′(x0))λ′(x0)

−1λ(z0) + H0j λ′(x0)

−1[x0, z0; λ]λ′(x0)−1λ(z0)‖

≤[∫ 1

0 ϕ((1− τ)g(‖x0 − s∗‖)‖x0 − s∗‖)dτ

1− ϕ0(g(‖x0 − s∗‖)‖x0 − s∗‖)

+

(ϕ0(‖x0 − s∗‖) + ϕ0(g(‖x0 − s∗‖)‖x0 − s∗‖)

) ∫ 10 ϕ1(τg(‖x0 − s∗‖)‖x0 − s∗‖)dτ

(1− ϕ0(g(‖x0 − s∗‖)‖x0 − s∗‖))(1− ϕ0(‖x0 − s∗‖))

+ 2ϕ3

((1+ψ1(‖x0−s∗‖)

)‖x0−s∗‖

2

) ∫ 10 ϕ1(τg(‖x0 − s∗‖)‖x0 − s∗‖)dτ

(1− ϕ0(‖x0 − s∗‖))2

+ϕ4(‖x0 − s∗‖)ϕ5(‖x0 − s∗‖, ‖y0 − s∗‖)

∫ 10 ϕ1(τg(‖x0 − s∗‖)‖x0 − s∗‖)dτ

(1− ϕ0(‖x0 − s∗‖))2

]‖z0 − s∗‖

≤ ψ2(‖x0 − s∗‖)‖x0 − s∗‖ ≤ ‖x0 − s∗‖.

(42)

Here, recalling that z0 = x0+y02 , we also used the estimates

‖z0 − s∗‖ =∥∥∥∥ x0 + y0

2− s∗

∥∥∥∥ ≤ 12(‖x0 − s∗‖+ ‖y0 − s∗‖)

≤ 12(1 + ψ1(‖x0 − s∗‖))‖x0 − s∗‖,

(43)

α = λ′(z0)−1 − λ′(x0)

−1 = λ′(z0)−1[(λ′(x0)− λ′(s∗)) + (λ′(s∗)− λ′(z0))

]λ′(x0)

−1,

β = (−2I + H0λ′(x0)−1[x0, z0; λ])λ′(x0)

−1,

andγ = −2I + (2I + H0

0)λ′(x0)

−1[x0, z0; λ]

= −2I + 2Iλ′(x0)−1[x0, z0; λ] + 2H0

0 λ′(x0)−1[x0, z0; λ]

= 2λ′(x0)−1([x0, z0; λ]− λ′(x0)) + H0

0 λ′(x0)−1[x0, z0; λ]

to obtain (41) and (42).

Hence, we arrived at the next following Theorem.

Theorem 4. Under the conditions (C), the conclusions of Theorem 1 hold for method (5).

3. Numerical Applications

We test the theoretical results on many examples. In addition, we use five examples and out ofthem: The first one is a counter example where the earlier results are applicable; the next three arereal life problems, e.g., a chemical engineering problem, an electron trajectory in the air gap amongtwo parallel surfaces problem, and integral equation of Hammerstein problem, which are displayed in

216


Examples 1–5. The last one compares favorably (5) to the other three methods. Moreover, the solutionto corresponding problem are also listed in the corresponding example which is correct up to 20significant digits. However, the desired roots are available up to several number of significant digits(minimum one thousand), but due to the page restriction only 30 significant digits are displayed.

We compare the four methods namely (2)–(5), denoted by NM, HM, JM, and BM, respectivelyon the basis of radii of convergence ball and the approximated computational order of convergence

ρ =log[‖x(j+1)−x(j)‖/‖x(j)−x(j−1)‖

]log[‖x(j)−x(j−1)‖/‖x(j−1)−x(j−2)‖

] , j = 2, 3, 4, ... (for the details please see Cordero and Torregrosa [5])

(ACOC). We have included the radii of ball convergence in the following Tables 1–6 except, the Table 4that belongs to the values of abscissas tj and weights wj. We use the Mathematica 9 programmingpackage with multiple precision arithmetic for computing work.

We choose in all examples H0(x) = 0 and H(x) = 2I, so ϕ2(t) = 1 and ϕ4(t) = 0. The divideddifference is [x, y; λ] =

∫ 10 λ′(y + θ(x− y))dθ. In addition, we choose the following stopping criteria

(i) ‖xj+1 − xj‖ < ε and (ii) ‖λ(xj)‖ < ε, where ε = 10−250.

Example 1. Set X = Y = R. We suggest a function λ on D = [− 1π , 2

π ] as

λ(x) =

{0, x = 0x5 sin (1/x) + x3 log(π2x2), x �= 0

.

But, λ′′′(x) is unbounded on Ω at x = 0. The solution of this problem is s∗ = 1π . The results in Nedzhibov [1],

Hueso et al. [2], Junjua et al. [3], and Behl et al. [4] cannot be utilized. In particular, conditions on the 5thderivative of λ or may be even higher are considered there to obtain the convergence of these methods. But, weneed conditions on λ′ according to our results. In additon, we can choose

H =80 + 16π + (π + 12 log 2)π2

2π + 1, ϕ1(t) = 1 + Ht, ϕ0(t) = ϕ(t) = Ht,

ϕ5(s, t) =12(

ϕ1(s) + ϕ1(t))

and ϕ3(t) =12

ϕ2(t).

The distinct radius of convergence, number of iterations n, and COC (ρ) are mentioned in Table 1.

Table 1. Comparison on the basis of different radius of convergence for Example 1.

Schemes R1 R2 R x0 n ρ

NM 0.011971 0.010253 0.010253 0.30831 4 4.0000HM 0.011971 0.01329 0.011971 0.32321 4 4.0000JM 0.011971 0.025483 0.011971 0.32521 4 4.0000BM 0 0 0 - - -

Equation (39) is violated with these choices of ϕi . This is the reason that R is zero in the method BM. Therefore,our results hold only, if x0 = s∗.

Example 2. The function

λ2(x) = x4 − 1.674− 7.79075x3 + 14.7445x2 + 2.511x. (44)

appears in the conversion to ammonia of hydrogen-nitrogen [6,7]. The function λ2 has 4 zeros, but we chooses∗ ≈ 3.9485424455620457727 + 0.3161235708970163733i. Moreover, we have

ϕ0(t) = ϕ(t) = 40.6469t, ϕ1(t) = 1 + 40.6469t, ϕ3(t) =12

ϕ2(t), and ϕ5(s, t) =12(

ϕ1(s) + ϕ1(t)).


217




NM 0.0098841 0.0048774 0.0048774 3.953 + 0.3197i 4 4.0000HM 0.0098841 0.016473 0.016473 3.9524 + 0.32i 4 4.0000JM 0.0098841 0.0059094 0.0059094 3.9436 + 0.3112i 4 4.0000BM 0 0 0 - - -


Example 3. An electron trajectory in the air gap among two parallel surfaces is formulated given as

x(t) =x0 +

(v0 + e

E0

mωsin(ωt0 + α)

)(t− t0) + e

E0

mω2

(cos(ωt + α) + sin(ω + α)

), (45)

where e, m, x0, v0, and E0 sin(ωt + α) are the charge, the mass of the electron at rest, the position, velocity ofthe electron at time t0, and the RF electric field among two surfaces, respectively. For particular values of theseparameters, the following simpler expression is provided:

f3(x) = x +π

4− 1

2cos(x). (46)

The solution of function f3 is s∗ ≈ −0.309093271541794952741986808924. Moreover, we have

ϕ(t) = ϕ0(t) = 0.5869t, ϕ1(t) = 1 + 0.5869t, ϕ3(t) =12

ϕ2(t) and ϕ5(s, t) =12(

ϕ1(s) + ϕ1(t)).




NM 0.678323 0.33473 0.33473 0.001 4 4.0000HM 0.678323 1.13054 0.678323 −0.579 4 4.0000JM 0.678323 0.40555 0.40555 0.091 5 4.0000BM 0 7.60065× 10−18 0 - - -


Example 4. Considering mixed Hammerstein integral equation Ortega and Rheinbolt [8], as

x(s) = 1 +15

∫ 1

0U(s, t)x(t)3dt, x ∈ C[0, 1], s, t ∈ [0, 1], (47)

where the kernel U is

U(s, t) =

{s(1− t), s ≤ t,

(1− s)t, t ≤ s.

We phrase (47) by using the Gauss-Legendre quadrature formula with∫ 1

0 φ(t)dt �10

∑k=1

wkφ(tk), where

tk and wk are the abscissas and weights respectively. Denoting the approximations of x(ti) with xi (i =

1, 2, 3, ..., 10), then we yield the following 8× 8 system of nonlinear equations

5xi − 5−10

∑k=1

aikx3k = 0, i = 1, 2, 3..., 10,

218


aik =

{wktk(1− ti), k ≤ i,

wkti(1− tk), i < k.

The values of tk and wk can be easily obtained from Gauss-Legendre quadrature formula when k = 8mentioned in Table 4.

Table 4. Values of abscissas tj and weights wj.

j tj wj

1 0.01304673574141413996101799 0.033335672154344068796784402 0.06746831665550774463395165 0.074725674575290296572888163 0.16029521585048779688283632 0.109543181257991021997767464 0.28330230293537640460036703 0.134633359654998177545613465 0.42556283050918439455758700 0.147762112357376435086946496 0.57443716949081560544241300 0.147762112357376435086946497 0.71669769706462359539963297 0.134633359654998177545613468 0.83970478414951220311716368 0.109543181257991021997767469 0.93253168334449225536604834 0.0747256745752902965728881610 0.98695326425858586003898201 0.03333567215434406879678440

The required approximate root is s∗ ≈ (1.001377, . . . , 1.006756, . . . , 1.014515, . . . , 1.021982, . . . ,1.026530, . . . , 1.026530, . . . , 1.021982, . . . , 1.014515, . . . , 1.006756, . . . , 1.001377, . . . )T. Moreover, we have

ϕ0(t) = ϕ(t) =320

t, ϕ1(t) = 1 +320

t, ϕ3(t) =12

ϕ2(t) and ϕ5(s, t) =12(

ϕ1(s) + ϕ1(t)).




NM 2.6667 1.3159 1.3159 (1,1,...,1) 4 4.0000HM 2.6667 4.4444 2.6667 (1.9,1.9,...,1.9) 5 4.0000JM 2.6667 1.5943 1.5943 (2.1,2.1,...,2.1) 5 4.0000BM 0 0 0 - - -


Example 5. We consider a boundary value problem from [8], which is defined as follows:

t′′ = 12

t3 + 3t′ − 32− x

+12

, t(0) = 0, t(1) = 1. (48)

We assume the following partition on [0, 1]

x0 = 0 < x1 < x2 < · · · < xj, where xj+1 = xj + h, h =1j.

We discretize this BVP (48) by

t′i ≈ti+1 − ti−1

2h, t′′i ≈

ti−1 − 2ti + ti+1

h2 , i = 1, 2, . . . , j− 1.

Then, we obtain a (k− 1)× (k− 1) order nonlinear system, given by

ti+1 − 2ti + ti−1 − h2

2t3i −

32− xi

h2 − 3ti+1 − ti−1

2h− 1

h2 = 0, i = 1, 2, . . . , j− 1,

219


where t0 = t(x0) = 0, t1 = t(x1), . . . , tj−1 = t(xj−1), tj = t(xj) = 1 and initial approximation

t(0)0 =(

12 , 1

2 , 12 , 1

2 , 12 , 1

2

)T. In particular, we choose k = 6 so that we can obtain a 5× 5 nonlinear system.

The required solution of this problem is

x ≈ (0.09029825 . . . , 0.1987214 . . . , 0.3314239 . . . , 0.4977132 . . . , 0.7123306 . . .

)T .


Table 6. Convergence behavior of distinct fourth-order methods for Example 5.

Methods j ‖F(x(j))‖ ‖x(j+1) − x(j)‖ ρ

MM1 8.1 (−6) 2.0 (−4)2 1.0 (−23) 3.1 (−23)3 9.1 (−95) 2.4 (−94)4 3.7 (−379) 9.0 (−379) 3.9996

HM1 7.8 (−6) 1.9 (−5)2 7.6 (−24) 2.4 (−23)3 2.7 (−95) 7.2 (−95)4 2.6 (−381) 6.3 (−381) 3.9997

JM1 7.8 (−6) 1.9 (−5)2 7.6 (−24) 2.4 (−23)3 2.7 (−95) 7.2 (−95)4 2.6 (−381) 6.3 (−381) 3.9997

BM1 7.2 (−6) 1.7 (−5)2 4.2 (−24) 1.3 (−23)3 1.9 (−96) 5.2 (−96)4 5.6 (−386) 1.4 (−385) 3.9997

4. Conclusions

The convergence order of iterative methods involves Taylor series, and the existence of highorder derivatives. Consequently, upper error bounds on ‖xj − s∗‖ and uniqueness results are notreported with this technique. Hence, the applicability of these methods is limited to functions withhigh order derivatives. To address these problems, we present local convergence results based onthe first derivative. Moreover, we compare methods (2)–(5). Notice that our convergence criteria aresufficient but not necessary. Therefore, if e.g., the radius of convergence for the method (5) is zero,that does not necessarily imply that the method does not converge for a particular numerical example.Our method can be adopted in order to expand the applicability of other methods in an analogousway.

Author Contributions: Both the authors have equal contribution for this paper.



References

1. Nedzhibov, G.H. A family of multi-point iterative methods for solving systems of nonlinear equations.Comput. Appl. Math. 2008, 222, 244–250. [CrossRef]

2. Hueso, J.L.; Martínez, E.; Teruel, C. Convergence, Efficiency and Dynamics of new fourth and sixth orderfamilies of iterative methods for nonlinear systems. Comp. Appl. Math. 2015, 275, 412–420. [CrossRef]

3. Junjua, M.; Akram, S.; Yasmin, N.; Zafar, F. A New Jarratt-Type Fourth-Order Method for Solving System ofNonlinear Equations and Applications. Appl. Math. 2015, 2015, 805278. [CrossRef]

4. Behl, R.; Cordero, A.; Torregrosa, J.R.; Alshomrani, A.S. New iterative methodsfor solving nonlinear problemswith one and several unknowns. Mathematics 2018, 6, 296. [CrossRef]

220


5. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas.Appl. Math. Comput. 2007, 190, 686–698. [CrossRef]

6. Balaji, G.V.; Seader, J.D. Application of interval Newton’s method to chemical engineering problems.Reliab. Comput. 1995, 1, 215–223. [CrossRef]

7. Shacham, M. An improved memory method for the solution of a nonlinear equation. Chem. Eng. Sci. 1989,44, 1495–1501. [CrossRef]

8. Ortega, J.M.; Rheinbolt, W.C. Iterative Solutions of Nonlinears Equations in Several Variables; Academic Press:Cambridge, MA, USA, 1970.


221

mathematics

Article

Extended Local Convergence for the CombinedNewton-Kurchatov Method Under the GeneralizedLipschitz Conditions

Ioannis K. Argyros 1 and Stepan Shakhno 2,*

1 Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA; [email protected] Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine* Correspondence: [email protected]

Received: 4 February 2019; Accepted: 20 February 2019; Published: 23 February 2019

Abstract: We present a local convergence of the combined Newton-Kurchatov method for solvingBanach space valued equations. The convergence criteria involve derivatives until the second andLipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notionof the restricted convergence region. These modifications of earlier conditions result in a tighterconvergence analysis and more precise information on the location of the solution. These advantagesare obtained under the same computational effort. Using illuminating examples, we further justifythe superiority of our new results over earlier ones.

Keywords: nonlinear equation; iterative process; non-differentiable operator; Lipschitz condition

MSC: 65H10; 65J15; 47H17

1. Introduction

Consider the nonlinear equation

F(x) + Q(x) = 0, (1)

where F is a Fréchet-differentiable nonlinear operator on an open convex subset D of a Banach spaceE1 with values in a Banach space E2, and Q : D → E2 is a continuous nonlinear operator.

Let x, y be two points of D. A linear operator from E1 into E2, denoted Q(x, y), which satisfiesthe condition

Q(x, y)(x− y) = Q(x)−Q(y) (2)

is called a divided difference of Q at points x and y.Let x, y, z be three points of D. A operator Q(x, y, z) will be called a divided difference of the

second order of the operator Q at the points x, y and z, if it satisfies the condition

Q(x, y, z)(y− z) = Q(x, y)−Q(x, z). (3)

A well-known simple difference method for solving nonlinear equations F(x) = 0 is theSecant method

xn+1 = xn − (F(xn−1, xn))−1F(xn), n = 0,1,2, . . . , (4)

where F(xn−1, xn) is a divided difference of the first order of F(x) and x0, x−1 are given.



Secant method for solving nonlinear operator equations in a Banach space was explored by theauthors [1–6] under the condition that the divided differences of a nonlinear operator F satisfy theLipschitz (Hölder) condition with constant L of type

‖F(x, y)− F(u, v)‖ ≤ L(‖x− u‖+ ‖y− v‖).

In [7] a one-point iterative Secant-type method with memory was psoposed.In [8,9] the Kurchatov method under the classical Lipschitz conditions for the divided differences

of the first and second order was explored and its quadratic convergence of it was determined.The iterative formula of Kurchatov method has the form [1,8–11]

xn+1 = xn − (F(2xn − xn−1, xn−1))−1F(xn), n = 0,1,2, . . . . (5)

Related articles but with stronger convergence criteria exist; see works of Argyros, Ezquerro,Hernandez, Rubio, Gutierrez, Wang, Li [1,12–15] and references therein.

In [14] which dealt with the study of the Newton method, it was proposed that thereare generalized Lipschitz conditions for the nonlinear operator, in which instead of constant L,some positive integrable function is used.

In our work [16], we introduced, for the first time, a similar generalized Lipschitz condition forthe operator of the first order divided difference, and under this condition, the convergence of theSecant method was studied and it was found that its convergence order is (1 +

√5)/2.

In [17], we introduced a generalized Lipschitz condition for the divided differences of the secondorder, and we have studied the local convergence of the Kurchatov method (5).

Note that in many papers, such as [3,18–21], the authors investigated the Secant and Secant-typemethods under the generalized conditions for the first divided differences of the form

‖(F(x, y)− F(u, v)))‖ ≤ ω(‖x− y‖, ‖u− v‖) ∀x, y, u, v ∈ D, (6)

where ω : R+ × R+ −→ R+ is continuous nondecreasing function in their two arguments. Underthese same conditions, in the work of Argyros [10], it was proven that there is a semi-local convergenceof the Kurchatov method and in [22] of Ren and Argyros the semi-local convergence of a combinedKurchatov method and Secant method was demonstrated. In both cases, only the linear convergenceof the methods is received.

We also refer the reader to the intersting paper by Donchev et al. [23], where several other relaxedLipschitz conditions are used in the setting of fixed points for these conditions. Clearly, our results canbe written in this setting too in an analogous way.

In [24], we first proposed and studied the local convergence of the combinedNewton-Kurchatov method

xn+1 = xn − (F′(xn) + Q(2xn − xn−1, xn−1))−1(F(xn) + Q(xn)), n = 0,1,2, . . . , (7)

where F′(u) is a Fréchet derivative, Q(u, v) is a divided difference of the first order, x0, x−1 are given,which is built on the basis of the mentioned Newton and Kurchatov methods. Semi-local convergenceof the method (7) under the classical Lipschitz conditions is studied in the mentioned article, but theconvergence only with the order (1 +

√5)/2 has been determined.

In [25], we studied the method (7) under relatively weak, generalized Lipschitz conditions for thederivatives and divided differences of nonlinear operators. Setting Q(x) ≡ 0, we receive the results forthe Newton method [14], and when F(x) ≡ 0 we got the known results for Kurchatov method [9,17].We proved the quadratic order of convergence of the method (7), which is higher than the convergenceorder (1 +

√5)/2 for the Newton–Secant method [1,26–28]

xn+1 = xn − (F′(xn) + Q(xn−1, xn))−1(F(xn) + Q(xn)), n = 0,1,2, . . . , . (8)

223


The results of the numerical study of the method (7) and other combined methods on the testproblems are provided in our works [24,28].

In this work, we continue to study a combined method (7) for solving nonlinear Equation (1),but with optimization considerations resulting in a tighter analysis than in [25].

The rest of the article is structured as follows: In Section 2, we present the local convergenceanalysis of the method (7) and the uniquness ball for solution of the equation. Section 3 contains theCorollaries of Theorems from Section 2. In Section 4, we provide the numerical example. The articleends with some conclusions.

2. Local Convergence of Newton-Kurchatov Method (7)

Let us denote B(x0, r) = {x : ‖x − x0‖ < r} an open ball of radius r > 0 with center at pointx0 ∈ D, B(x0, r) ⊂ D.

Condition on the divided difference operator Q(x, y)

‖Q(x, y)−Q(u, v)‖ ≤ L(‖x− u‖+ ‖y− v‖) ∀x, y, u, v ∈ D (9)

is called Lipschitz condition in domain D with constant L > 0. If the condition is being fulfilled

‖Q(x, y)−Q′(x0)‖ ≤ L(‖x− x0‖+ ‖y− x0‖) ∀x, y ∈ B(x0, r), (10)

then we call it the center Lipschitz condition in the ball B(x0, r) with constant L.However, L in Lipschitz conditions can be not a constant, and can be a positive integrable function.

In this case, if for x∗ ∈ D inverse operator [F′(x∗)]−1 exists, then the conditions (9) and (10) for x0 = x∗can be replaced respectively for

‖Q′(x∗)−1(Q(x, y)−Q(u, v)))‖ ≤∫ ‖x−y‖+‖u−v‖

0L(t)dt ∀x, y, u, v ∈ D (11)

and

‖Q′(x∗)−1(Q(x, y)−Q′(x∗))‖ ≤∫ ‖x−x∗‖+‖y−x∗‖

0L(t)dt ∀x, y ∈ B(x∗, r). (12)

SimultaneouslyLipschitz conditions (11) and (12) are called generalized Lipschitz conditions or Lipschitz

conditions with the L average.Similarly, we introduce the generalized Lipschitz condition for the divided difference of the

second order

‖Q′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤∫ ‖u−v‖

0N(t)dt ∀x, y, u, v ∈ B(x∗, r), (13)

where N is a positive integrable function.

Remark 1. Note than the operator F is Fréchet differentiable on D when the Lipschitz conditions (9) or (11) arefulfilled ∀x, y, u, v ∈ D (the divided differences F(x, y) are Lipschitz continuous on D) and F(x, x) = F′(x)∀x ∈ D [29].

Suppose that equation

∫ r

0L0

1(u)du +∫ 2r

0L0

2(u)du + 2r∫ 2r

0N0(u)du = 1.

has at least one positive solution. Denote by r0 the smallest such solution. Set D0 = D ∩ B(x∗, r0)

224


The radius of the convergence ball and the convergence order of the combined Newton–Kurchatovmethod (7) are determined in next theorem.

Theorem 1. Let F and Q be continuous nonlinear operators defined in open convex domain D of a Banachspace E1 with values in the Banach space E2. Let us suppose, that: (1) H(x) ≡ F(x) + Q(x) = 0 has a solutionx∗ ∈ D, for which there exists a Fréchet derivative H′(x∗) and it is invertible; (2) F has the Fréchet derivative ofthe first order, and Q has divided differences of the first and second order on B(x∗, 3r) ⊂ D, so that for eachx, y, u, v ∈ D

‖H′(x∗)−1(F′(x)− F′(x∗))‖ ≤∫ ρ(x)

0L0

1(u)du, (14)

‖H′(x∗)−1(Q(x, y)−Q(x∗, x∗))‖ ≤∫ ‖x−x∗‖+‖y−x∗‖

0L0

2(t)dt, (15)

‖H′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤∫ ‖u−v‖

0N0(t)dt, (16)

and for each x, y, u, v ∈ D0

‖H′(x∗)−1(F′(x)− F′(xθ))‖ ≤∫ ρ(x)

θρ(x)L1(u)du, 0 ≤ τ ≤ 1, (17)

‖H′(x∗)−1(Q(x, y)−Q(u, v))‖ ≤∫ ‖x−u‖+‖y−v‖

0L2(t)dt, (18)

‖H′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤∫ ‖u−v‖

0N(t)dt, (19)

where xθ = x∗ + θ(x− x∗), �(x) = ‖x− x∗‖, L01, L0

2, N0 L1, L2 and N are positive nondecreasing integrablefunctions and r > 0 satisfies the equation

1r∫ r

0 L1(u)udu +∫ r

0 L2(u)du + 2r∫ 2r

0 N(u)du

1−( ∫ r

0 L01(u)du +

∫ 2r0 L0

2(u)du + 2r∫ 2r

0 N0(u)du) = 1. (20)

Then for all x0, x−1 ∈ B(x∗, r) the iterative method (7) is well defined and the generated by it sequence{xn}n≥0, which belongs to B(x∗, r), converges to x∗ and satisfies the inequality

‖xn+1 − x∗‖ ≤ en :=1

ρ(xn)

∫ ρ(xn)0 L1(u)udu +

∫ ρ(xn)0 L2(u)du +

∫ ‖xn−xn−1‖0 N(u)du‖xn − xn−1‖

1−( ∫ ρ(xn)

0 L01(u)du +

∫ 2ρ(xn)0 L0

2(u)du +∫ ‖xn−xn−1‖

0 N0(u)du‖xn − xn−1‖)‖xn − x∗‖. (21)

Proof. First we show that f (t) =1t2

∫ t

0L1(u)udu, g(t) =

1t

∫ t

0L2(u)du, h(t) =

1t

∫ t

0N(u)du, f0(t) =

1t2

∫ t

0L0

1(u)udu, g0(t) =1t

∫ t

0L0

2(u)du, h0(t) =1t

∫ t

0N0(u)du monotonically nondecreasing with

respect to t. Indeed, under the monotony of L1, L2, N we have( 1t22

∫ t2

0− 1

t21

∫ t1

0

)L1(u)udu =

( 1t22

∫ t2

t1

+( 1

t22− 1

t21

) ∫ t1

0

)L1(u)udu ≥

≥ L(t1)( 1

t22

∫ t2

t1

+( 1

t22− 1

t21

) ∫ t1

0

)udu = L1(t1)

( 1t22

∫ t2

0− 1

t21

∫ t1

0

)udu = 0,

225


( 1t2

∫ t2

0− 1

t1

∫ t1

0

)L2(u)du =

( 1t2

∫ t2

t1

+( 1

t2− 1

t1

) ∫ t1

0

)L2(u)du ≥

≥ L2(t1)( 1

t2

∫ t2

t1

+( 1

t2− 1

t1

) ∫ t1

0

)du = L2(t1)

( t2 − t1

t2+ t1

( 1t2− 1

t1

))= 0

for 0 < t1 < t2. So, f (t), g(t) are nondecreasing with respect to t. Similarly we get for h(t), f0(t), g0(t)and h0(t).

We denote by An linear operator An = F′(xn)+Q(2xn− xn−1, xn−1). Easy to see that if xn, xn−1 ∈B(x∗, r), then 2xn − xn−1, xn−1 ∈ B(x∗, 3r). Then An is invertible and the inequality holds

‖A−1n H

′(x∗)‖ = ‖[I − (I − H

′(x∗)−1 An)]−1‖ ≤

≤(

1−( ∫ ρ(xn)

0L0

1(u)du +∫ 2ρ(xn)

0L0

2(u)du +∫ ‖xn−xn−1‖

0N0(u)du‖xn − xn−1‖

))−1.

(22)

Indeed from the formulas (14)–(16) we get

‖I − H′(x∗)−1 An‖ = ‖H

′(x∗)−1(F′(x∗)− F′(xn) + Q(x∗, x∗)−Q(xn, xn)+

+Q(xn, xn)−Q(2xn − xn−1, xn−1)‖) ≤∫ ρ(xn)

0L0

1(u)du + ‖H′(x∗)−1(Q(x∗, x∗)−

−Q(xn, xn) + Q(xn, xn)−Q(xn, xn−1) + Q(xn, xn−1)−Q(2xn − xn−1, xn−1))‖ ≤

≤∫ ρ(xn)

0L0

1(u)du +∫ 2ρ(xn)

0L0

2(u)du+

+‖H′(x∗)−1(Q(xn, xn−1, xn)−Q(2xn − xn−1, xn−1, xn))(xn − xn−1)‖ ≤

≤∫ ρ(xn)

0L0

1(u)du +∫ 2ρ(xn)

0L0

2(u)du +∫ ‖xn−xn−1‖

0N0(u)du‖xn − xn−1‖.

From the definition r0 (20), we get

∫ r0

0L1(u)du +

∫ 2r0

0L2(u)du + 2r

∫ 2r0

0N(u)du < 1, (23)

since r < r0.Using the Banach theorem on inverse operator [30], we get formula (22). Then we can write

‖xn+1 − x∗‖ = ‖xn − x∗ − A−1n (F(xn)− F(x∗) + Q(xn)−Q(x∗))‖ =

= ‖ − A−1n (

∫ 1

0(F′(xτ

n)− F′(xn))dτ + Q(xn, x∗)−Q(2xn − xn−1, xn−1))(xn − x∗)‖ ≤

≤ ‖A−1n H

′(x∗)‖(‖H

′(x∗)−1

∫ 1

0

∫ ρ(xn)

τρ(xn)L1(u)dudτ + ‖H

′(x∗)−1(+Q(xn, x∗)−

−Q(2xn − xn−1, xn−1))‖)‖xn − x∗‖.

(24)

226


According to the condition (17)–(19) of the theorem we get

‖H′(x∗)−1(

∫ 1

0

∫ ρ(xn)

τρ(xn)L1(u)dudτ + Q(xn, x∗)− An)‖ =

=1

ρ(xn)

∫ ρ(xn)

0L1(u)udu + ‖H′(x∗)−1(Q(xn, x∗)−Q(xn, xn)+

+Q(xn, xn)−Q(xn, xn−1) + Q(xn, xn−1)−Q(2xn − xn−1, xn−1))‖ ≤

≤ 1ρ(xn)

∫ ρ(xn)

0L1(u)udu + ‖H′(x∗)−1(Q(xn, x∗)−Q(xn, xn))‖+

+‖H′(x∗)−1(Q(xn, xn−1, xn)−Q(2xn − xn−1, xn−1, xn))(xn − xn−1)‖ ≤

≤ 1ρ(xn)

∫ ρ(xn)

0L1(u)udu +

∫ ρ(xn)

0L2(u)du +

∫ ‖xn−xn−1‖

0N(u)du‖xn − xn−1‖.

From (22) and (24) shows that fulfills (21). Then from (21) and (20) we get

‖xn+1 − x∗‖ < ‖xn − x∗‖ < ... < max{‖x0 − x∗‖, ‖x−1 − x∗‖} < r.

Therefore, the iterative process (5) is correctly defined and the sequence that it generates belongsto B(x∗, r). From the last inequality and estimates (21) we get lim

n→∞‖xn − x∗‖ = 0. Since the sequence

{xn}n≥0 converges to x∗, then

‖xn − xn−1‖ ≤ ‖xn − x∗‖+ ‖xn−1 − x∗‖ ≤ 2‖xn−1 − x∗‖

and limn→∞

‖xn − xn−1‖ = 0.

Corollary 1. The order of convergence of the iterative procedure (7) is quadratic.

Proof. Let us denote ρmax = max{ρ(x0), ρ(x−1)}. Since g(t) and h(t) are monotonicallynondecreasing, then with taking into account the expressions

1ρ(xn)

∫ ρ(xn)

0L1(u)udu =

∫ ρ(xn)0 L1(u)uduρ(xn))

(ρ(xn))2 ≤∫ ρmax

0 L1(u)uduρ(xn)

(ρmax)2 =: A1ρ(xn),

∫ ρ(xn)

0L2(u)du =

∫ ρ(xn)0 L2(u)duρ(xn)

ρ(xn)≤∫ ρmax

0 L2(u)duρ(xn)

ρmax=: A2ρ(xn),

∫ ‖xn−xn−1‖

0N(u)du =

∫ ‖xn−xn−1‖0 N(u)du‖xn − xn−1‖

‖xn − xn−1‖ <

<

∫ ‖x0−x−1‖0 N(u)du‖xn − xn−1‖

‖x0 − x−1‖ =: A3‖xn − xn−1‖

and (1−

( ∫ ρ(xn)

0L0

1(u)du + 2∫ ρ(xn)

0L0

2(u)du +∫ ‖xn−xn−1‖

0N0(u)du‖xn − xn−1‖

))−1<

<(

1−( ∫ ρmax

0L0

1(u)du + 2∫ ρmax

0L0

2(u)du +∫ ‖x0−x−1‖

0N0(u)du‖x0 − x−1‖

))−1=: A4,

227


from the inequality (21) follows

‖xn+1 − x∗‖ ≤ A4(A1ρ(xn) + A2ρ(xn) + A3‖xn − xn−1‖2)‖xn − x∗‖.

or‖xn+1 − x∗‖ ≤ C3‖xn − x∗‖2 + C4‖xn − xn−1‖2‖xn − x∗‖. (25)

Here Ak, k = 1, ..., 4, C3, C4 are some positive constants.Assume that the order of convergence of the iterative process (7) is not lower 2, therefore there

exist C5 ≥ 0 and N > 0, that for all n ≥ N the inequality holds

‖xn − x∗‖ ≥ C5‖xn−1 − x∗‖2.

Since‖xn − xn−1‖2 ≤ (‖xn − x∗‖+ ‖xn−1 − x∗‖)2 ≤ 4‖xn−1 − x∗‖2,

then from (44) we get

‖xn+1 − x∗‖ ≤ C3‖xn − x∗‖2 + 4C4‖xn−1 − x∗‖2‖xn − x∗‖

≤ (C3 + 4C4/C5)‖xn − x∗‖2 = C6‖xn − x∗‖2.(26)

inequality (26) means that the order of convergence is not lower than 2. Thus, the convergencerate of sequence {xn}n≥0 to x∗ is quadratic.

3. Uniqueness Ball of the Solution

The next theorem determines the ball of uniqueness of the solution x∗ of (1) in B(x∗, r).

Theorem 2. Let us assume that: (1) H(x) ≡ F(x) + Q(x) = 0 has a solution x∗ ∈ D, in which there exists aFréchet derivative H′(x∗) and it is invertible; (2) F has a continuous Frechet derivative in B(x∗, r), F′ satisfiesthe generalized Lipschitz condition

‖H′(x∗)−1(F′(x)− F′(x∗))‖ ≤∫ ρ(x)

0L0

1(u)du ∀x ∈ B(x∗, r),

the divided difference Q(x, y) satisfies the generalized Lipschitz condition

‖H′(x∗)−1(Q(x, x∗)− G′(x∗))‖ ≤∫ ρ(x)

0L0

2(u)du ∀x ∈ B(x∗, r),

where L1 and L2 are positive integrable functions. Let r > 0 satisfy

1r

∫ r

0(r− u)L0

1(u)du +∫ r

0L0

2(u)du ≤ 1.

Then the equation H(x) = 0 has a unique solution x∗ in B(x∗, r).

Proof analogous to [27,31].

4. Corollaries

In the study of iterative methods, the traditional assumption is that the derivatives and/or thedivided differences satisfy the classical Lipschitz conditions. Assuming that L1, L2 and N are constants,we get from Theorems 1 and 2 important corollaries, which are of interest.

228


Corollary 2. Let us assume that: (1) H(x) ≡ F(x) + Q(x) = 0 has a solution x∗ ∈ D, in which there existsFréchet derivative H′(x∗) and it is invertible; (2) F has a continuous Fréchet derivative and Q has divideddifferences of the first and second order Q(x, y) and Q(x, y, z) in B(x∗, 3r) ⊂ D, which satisfy the Lipschitzconditions for each x, y, u, v ∈ D

‖H′(x∗)−1(F′(x)− F′(x∗)‖ ≤ L01‖x− x∗‖,

‖H′(x∗)−1(Q(x, y)−Q(u, v))‖ ≤ L02(‖x− u‖+ ‖y− v‖),

for x, y, u, v ∈ D0

‖H′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤ N0‖u− v‖,

‖H′(x∗)−1(F′(x)− F′(x∗ + τ(x− x∗))‖ ≤ (1− τ)L1‖x− x∗‖,

‖H′(x∗)−1(Q(x, y)−Q(u, v))‖ ≤ L2(‖x− u‖+ ‖y− v‖),‖H′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤ N‖u− v‖,

where L01, L0

2, N0, L1, L2 and N are positive numbers,

r0 =2

L01 + 2L0

2 +√(L0

1 + 2L02)

2 + 16N0

,

and r is the positive root of the equation

L1r/2 + L2r + 4Nr2

1− L01r− 2L0

2r− 4N0r2= 1.

Then Newton-Kurchatov method (5) converges for all x−1, x0 ∈ B(x∗, r) and there fulfills

‖xn+1 − x∗‖ ≤ (L1/2 + L2)‖xn − x∗‖+ N‖xn − xn−1‖2

1−(

L01 + 2L0

2‖xn − x∗‖+ N0‖xn − xn−1‖2) .

Moreover, r is the best of all possible.

Note that value of r =2

3Limproves r =

23L1

1for Newton method for solving equation F(x) =

0 [14,32,33], and with r = 2/(3L2 +√

9L22 + 32N) improves r = 2/(3L1

2 +√

9(L12)

2 + 32N1) forKurchatov method for solving the equation Q(x) = 0, as derived in [8].

Corollary 3. Suppose that: (1) H(x) ≡ F(x) + Q(x) = 0 has a solution x∗ ∈ D, in which there exists theFréchet derivative H′(x∗) and it is invertible; (2) F has continuous derivative and Q has divided differenceQ(x, x∗) in B(x∗, r) ⊂ D, which satisfy the Lipschitz conditions

‖H′(x∗)−1(F′(x)− F′(x∗))‖ ≤ L01‖x− x∗‖,

‖H′(x∗)−1(Q(x, x∗)− G′(x∗))‖ ≤ L02‖x− x∗‖

for all x ∈ B(x∗, r), where L01 and L0

2 are positive numbers and r =2

L01 + 2L0

2. Then x∗ is the only solution in

B(x∗, r) of H(x) = 0 , r does not depend on F and Q and is the best choice.

229


Note that the resulting radius of the uniqueness ball of the solution r =2L1

improves r =2L1

1for

Newton method for solving the equation F(x) = 0 [14] and r =1L2

improves r =1L1

2for Kurchatov

method for solving the equation Q(x) = 0 [8]. (See also the numerical examples).

Remark 2. We compare the results in [25] with the new results in this article. In order to do this, let us considerthe conditions given in [25] corresponding to our conditions (15)–(17):

For each x, y, u, v ∈ D

‖H′(x∗)−1(F′(x)− F′(xθ))‖ ≤∫ ρ(x)

θρ(x)L1

1(u)du, 0 ≤ θ ≤ 1, (27)

‖H′(x∗)−1(Q(x, y)−Q(u, v))‖ ≤∫ ‖x−u‖+‖y−v‖

0L1

2(t)dt, (28)

‖H′(x∗)−1(Q(u, x, y)−Q(v, x, y))‖ ≤∫ ‖u−v‖

0N1(t)dt, (29)

1r∫ r

0 L11(u)udu +

∫ r0 L1

2(u)du + 2r∫ 2r

0 N1(u)du

1−( ∫ r

0 L11(u)du +

∫ 2r0 L1

2(u)du + 2r∫ 2r

0 N11(u)du) = 1, (30)

‖xn+1 − xn‖ ≤ en. (31)

It follows from (14)–(16), (17)–(19), (27)–(29), that

L01(t) ≤ L1

1(t), (32)

L1(t) ≤ L11(t), (33)

L02(t) ≤ L1

2(t), (34)

L2(t) ≤ L12(t), (35)

N0(t) ≤ N1(t), (36)

N(t) ≤ N1(t), (37)

leading tor ≤ r, (38)

en ≤ en, (39)

Al ≤ Al , l = 1, 2, 3, 4, (40)

Cl ≤ Cl , l = 1, 2, 3, 4, 6 (41)

andC5 ≥ C5, (42)

en :=1

ρ(xn)

∫ ρ(xn)0 L1

1(u)udu +∫ ρ(xn)

0 L12(u)du +

∫ ‖xn−xn−1‖0 N1(u)du‖xn − xn−1‖

1−( ∫ ρ(xn)

0 L11(u)du +

∫ 2ρ(xn)0 L1

2(u)du +∫ ‖xn−xn−1‖

0 N1(u)du‖xn − xn−1‖)‖xn − x∗‖, (43)

‖xn+1 − x∗‖ ≤ A4(A1ρ(xn) + A2ρ(xn) + A3‖xn − xn−1‖2)‖xn − x∗‖,

230


or

‖xn+1 − x∗‖ ≤ C3‖xn − x∗‖2 + C4‖xn − xn−1‖2‖xn − x∗‖, (44)

or

‖xn+1 − x∗‖ ≤ C6‖xn − x∗‖2

with

C6 = (C3 + 4C4/C5)

for some‖xn − x∗‖ ≥ C5‖xn−1 − x∗‖2.

Hence, we obtain the impovements:

(1) At least as many initial choices x−1, x0 as before.(2) At least as few iterations than before to obtain a predetermined error accuracy.(3) At least as precice information on the location of the solution as before.

Moreover, if any of (32)–(37) holds as a strict inequality, then so do (38)–(42). Furthermore,we notice that these improvements are found using the same information, since the functions L0

1, L02, N0,

L1, L2, N are special cases of functions L11, L1

2, N1 used in [25]. Finally, if G = 0 or F = 0, we obtain theresults for Newton’s method or the Kurchatov method as special cases. Clearly, the results for thesemethods are also improved. Our technique can also be used to improve the results of other iterativemethods in an analogous way.


Example 1. Let E1 = E2 = R3 and Ω = S(x∗, 1). Define functions F and Q for v = (v1, v2, v3)T on Ω by

F(v) =(ev1 − 1, e−1

2 v22 + v2, v3

)T ,Q(v) =

(|v1|, |v2|, |v2|, | sin(v3)|)T (45)

F′(v) = diag(

ev1 , (e− 1)v2 + 1, 1)

,

Q(v, v) = diag( |v1|−|v1|

v1−v1, |v2|−|v2|

v2−v2, | sin(v3)|−| sin(v3)|

v3−v3

) (46)

Choose:

H(x) = F(x) + Q(x),

‖H′(x∗)−1‖ = 1, L01 = e− 1, L0

2 = 1, N0 =12

,

L1 = e1

e−1 , L2 = 1, N =12

,

L11 = e, L1

2 = 1, N1 =12

.

Then compute:r using (20), r = 0.1599;r using (30), r = 0.1315.Also, r < r.Notice that L0

1 < L1 < L11, so the improve ments stated in Remark 1 hold.

231


6. Conclusions

In [1,8,34], we studied the local convergence of Secant and Kurchatov methods in the case offulfilment of Lipschitz conditions for the divided differences, which hold for some Lipschitz constants.In [14], the convergence of the Newton method is shown for the generalized Lipschitz conditions forthe Fréchet derivative of the first order. We explored the local convergence of the Newton-Kurchatovmethod under the generalized Lipschitz conditions for Fréchet derivative of a differentiable part of theoperator and the divided differences of the nondifferentiable part. Our results contain known parts aspartial cases.

By using our idea of restricted convergence regions, we find tighter Lipschitz constants leading toa finer local convergence analysis of method (7) and its special cases compared to in [25].

Author Contributions: All authors contributed equally and significantly to writing this article. All authors readand approved the final manuscript.


Acknowledgments: The authors would like to express their sincere gratitude to the referees for their valuablecomments which have significantly improved the presentation of this paper.


References

1. Argyros, I.K. Convergence and Applications of Newton-Type Iterations; Springer: New York, NY, USA, 2008.2. Hernandez, M.A.; Rubio, M.J. The Secant method and divided differences Hölder continuous.

Appl. Math. Comput. 2001, 124, 139–149. [CrossRef]3. Hernandez, M.A.; Rubio, M.J. The Secant method for nondifferentiable operators. Appl. Math. Lett. 2002, 15,

395–399. [CrossRef]4. Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press:

New York, NY, USA, 1970.5. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall, Inc.: Englewood Cliffs, NJ, USA, 1964.6. Shakhno, S.M. Application of nonlinear majorants for intestigation of the secant method for solving nonlinear

equations. Matematychni Studii 2004, 22, 79–86.7. Ezquerro, J.A.; Grau-Sánchez, M.; Hernández, M.A. Solving non-differentiable equations by a new one-point

iterative method with memory. J. Complex. 2012, 28, 48–58. [CrossRef]8. Shakhno, S.M. On a Kurchatov’s method of linear interpolation for solving nonlinear equations. Proc. Appl.

Math. Mech. 2004, 4, 650–651. [CrossRef]9. Shakhno, S.M. About the difference method with quadratic convergence for solving nonlinear operator

equations. Matematychni Studii 2006, 26, 105–110. (In Ukrainian)10. Argyros, I.K. A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations. J. Math.

Anal. Appl. 2007, 332, 97–108. [CrossRef]11. Kurchatov, V.A. On a method of linear interpolation for the solution of functional equations. Dokl. Akad.

Nauk SSSR 1971, 198, 524–526. (In Russian); translation in Soviet Math. Dokl. 1971, 12, 835–838.12. Ezquerro, J.A.; Hernández, M. Generalized differentiability conditions for Newton’s method. IMA J. Numer.

Anal. 2002, 22, 187–205. [CrossRef]13. Gutiérrez, J.M.; Hernández, M.A. Newton’s method under weak Kantorovich conditions. IMA J. Numer. Anal.

2000, 20, 521–532. [CrossRef]14. Wang, X.H. Convergence of Newton’s method and uniqueness of the solution of equations in Banach space.

IMA J. Numer. Anal. 2000, 20, 123–134. [CrossRef]15. Wang, X.H.; Li, C. Local and global behavior for algorithms of solving equations. Chin. Sci. Bull. 2001, 46,

444–451. [CrossRef]16. Shakhno, S.M. On the secant method under generalized Lipschitz conditions for the divided difference

operator. Proc. Appl. Math. Mech. 2007, 7, 2060083–2060084. [CrossRef]17. Shakhno, S.M. Method of linear interpolation of Kurchatov under generalized Lipschitz conditions for divided

differences of first and second order. Visnyk Lviv. Univ. Ser. Mech. Math. 2012, 77, 235–242. (In Ukrainian)

232


18. Amat, S. On the local convergence of Secant-type methods. Intern. J. Comput. Math. 2004, 81, 1153–1161.[CrossRef]

19. Amat, S.; Busquier, S. On a higher order Secant method. Appl. Math. Comput. 2003, 141, 321–329. [CrossRef]20. Argyros, I.K.; Ezquerro, J.A.; Gutiérrez, J.M.; Hernández, M.A.; Hilout, S. Chebyshev-Secant type methods for

non-differentiable operator. Milan J. Math. 2013, 81, 25–35.21. Ren, H. New sufficient convergence conditions of the Secant method nondifferentiable operators.

Appl. Math. Comput. 2006, 182, 1255–1259. [CrossRef]22. Ren, H.; Argyros, I.K. A new semilocal convergence theorem with nondifferentiable operators. J. Appl.

Math. Comput. 2010, 34, 39–46. [CrossRef]23. Donchev, T.; Farkhi, E.; Reich, S. Fixed set iterations for relaxed Lipschitz multimaps. Nonlinear Anal. 2003, 53,

997–1015. [CrossRef]24. Shakhno, S.M.; Yarmola, H.P. Two-point method for solving nonlinear equation with nondifferentiable

operator. Matematychni Studii. 2011, 36, 213–220. (In Ukrainian)25. Shakhno, S.M. Combined Newton-Kurchatov method under the generalized Lipschitz conditions for the

derivatives and divided differences. J. Numer. Appl. Math. 2015, 2, 78–89.26. Catinas, E. On some iterative methods for solving nonlinear equations. Revue d’Analyse Numérique et de Théorie

de l’Approximation 1994, 23, 47–53.27. Shakhno, S. Convergence of combined Newton-Secant method and uniqueness of the solution of nonlinear

equations. Visnyk Ternopil Nat. Tech. Univ. 2013, 69, 242–252. (In Ukrainian)28. Shakhno, S.M.; Mel’nyk, I.V.; Yarmola, H.P. Analysis of convergence of a combined method for the solution of

nonlinear equations. J. Math. Sci. 2014, 201, 32–43. [CrossRef]29. Argyros, I. K. On the secant method. Publ. Math. Debr. 1993, 43, 233–238.30. Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Pergamon Press: Oxford, UK, 1982.31. Shakhno, S.M. Convergence of the two-step combined method and uniqueness of the solution of nonlinear

operator equations. J. Comput. Appl. Math. 2014, 261, 378–386. [CrossRef]32. Potra, F.A. On an iterative algorithm of order 1.839... for solving nonlinear operator equations. Numer. Funct.

Anal. Optim. 1985, 7, 75–106. [CrossRef]33. Traub, J.F.; Wozniakowski, H. Convergence and complexity of Newton iteration for operator equations.

J. Assoc. Comput. Mach. 1979, 26, 250–258. [CrossRef]34. Hernandez, M.A.; Rubio, M.J. A uniparametric family of iterative processes for solving nondifferentiable

equations. J. Math. Anal. Appl. 2002, 275, 821–834. [CrossRef]


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mathematics

Article

Study of a High Order Family: Local Convergenceand Dynamics

Cristina Amorós 1, Ioannis K. Argyros 2, Ruben González 1 , Á. Alberto Magreñán 3,

Lara Orcos 4 and Íñigo Sarría 1,*

1 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño, Spain;[email protected] (C.A.); [email protected] (R.G.)

2 Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA; [email protected] Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain;

[email protected] Facultad de Educación, Universidad Internacional de La Rioja, 26006 Logroño, Spain; [email protected]* Correspondence: [email protected]

Received: 10 December 2018; Accepted: 25 February 2019; Published: 28 February 2019

Abstract: The study of the dynamics and the analysis of local convergence of an iterative method,when approximating a locally unique solution of a nonlinear equation, is presented in this article.We obtain convergence using a center-Lipschitz condition where the ball radii are greater thanprevious studies. We investigate the dynamics of the method. To validate the theoretical resultsobtained, a real-world application related to chemistry is provided.

Keywords: high order; sixteenth order convergence method; local convergence; dynamics

1. Introduction

A well known problem is that of approximating a locally unique solution x∗ of equation

F(x) = 0, (1)

where F is a differentiable function defined on a nonempty convex subset D of S with values in Ω,where Ω can be R or C. In this article, we are going to deal with it.

Mathematics is always changing and the way we teach it also changes as it is presented in [1,2].In the literature [3–8], we can find many problems in engineering and applied sciences that can besolved by finding solutions of equations in a way such as (1). Finding exact solutions for this typeof equation is not easy. Only in a few special cases can we find the solutions of these equations inclosed form. We must look for other ways to find solutions to these equations. Normally we resortto iterative methods to be able to find solutions. Once we propose to find the solution iteratively,it is mandatory to study the convergence of the method. This convergence is usually seen in twodifferent ways, which gives rise to two different categories, the semilocal convergence analysis andthe local convergence analysis. The first of these, the semilocal convergence analysis, is based oninformation around an initial point, which will provide us with criteria that will ensure the convergenceof an iteration procedure. On the other hand, the local convergence analysis is generally based oninformation about a solution to find values of the calculated radii of the convergence balls. The localresults obtained are fundamental since they provide the degree of difficulty to choose the initial points.

We must also deal with the domain of convergence in the study of iterative methods. Normally,the convergence domain is very small and it is necessary to be able to extend this convergence domainwithout adding any additional hypothesis. Another important problem is finding more accurateestimates of error in distances. ‖xn+1 − xn‖, ‖xn − x∗‖. Therefore, to extend the domain without the



need for additional hypotheses and to find more precise estimates of the error committed, in additionto the study of dynamic behavior, will be our objectives in this work.

The iterative methods can be applied to polynomials, and the dynamic properties related to thismethod will give us important information about its stability and reliability. Recently in some studies,authors such as Amat et al. [9–11], Chun et al. [12], Gutiérrez et al. [13], Magreñán [14–16], and manyothers [8,13,17–30] have studied interesting dynamic planes, including periodic behavior and otheranomalies detected. For all the above, in this article, we are going to study the parameter spacesassociated with a family of iterative methods, which will allow us to distinguish between bad andgood methods, always speaking in terms of their numerical properties.

We present the dynamics and the local convergence of the four step method defined for eachn = 0, 1, 2, . . . by

yn = xn − αF′(xn)−1F(xn)

zn = yn − C1(xn)F′(xn)−1F(yn)

vn = zn − C2(xn)F′(xn)−1F(zn)

xn+1 = zn − C3(xn)F′(xn)−1F(vn),

(2)

where α ∈ R is a parameter, x0 is an initial point and Ci : R → R, i = 1, 2, 3 are continuousfunctions given. Numerous methods of more than one step are particular cases of the previousmethod (2). For example, for certain values of the parameters this family can be reduced to:

• Artidiello et al. method [31]• Petkovic et al. method [32]• Kung-Traub method [29]• Fourth order King family• Fourth order method given by Zhao et al. in [33]• Eighth order method studied by Dzunic et al. [34].

It should be noted that to demonstrate the convergence of all methods after the method (2),in all cases Taylor expansions have been used as well as hypotheses involving derivatives of ordergreater than one, usually the third derivative or greater. However, in these methods only the firstderivative appears. In this article we will perform the analysis of local convergence of the method (2)using hypotheses that involve only the first derivative of the function F. In this way we save thetedious calculation of the successive derivatives (in this case the second and third derivatives) in eachstep. The order of convergence (COC) is found using and an approximation of the COC (ACOC) usingthat do not require the usage of derivatives of order higher than one (see Remark 1). Our objective willalso be able to provide a computable radius of convergence and error estimates based on the Lipschitzconstants.

We must also realize that there are a lot of iterative methods to approximate solutions of nonlinearequations defined in R or C [32,35–38]. These studies show that if the initial point x0 is close enoughto the solution x∗, the sequence {xn} converges to x∗. However, from the initial estimate, how close tothe solution x∗ should it be? In these cases, the local results do not provide us with information aboutthe radius of the convergence ball for the corresponding method. We will approach this question forthe method (2) in Section 2. Similarly, we can use the same technique with other different methods.

2. Method’s Local Convergence

Let us define, respectively, U(v, ρ) and U(v, ρ) as open and closed balls in S, of radius ρ > 0 andwith center v ∈ Ω.

To study the analysis of local convergence of the method (2), we are going to define a series ofconditions that we will name (C):

(C1) F : D ⊂ Ω → Ω is a differentiable function.

235


We know that exist a constant x∗ ∈ D, L0 > 0, such that for each x ∈ D is fulfilled(C2) F(x∗) = 0, F′(x∗) �= 0.(C3) ‖F′(x∗)−1(F′(x)− F′(x∗))‖ ≤ L0‖x− x∗‖

Let D0 := D ∩U(x∗, 1L0

). There exist constants L > 0, M ≥ 1 such that for each , y ∈ D0

(C4) ‖F′(x∗)−1(F′(x)− F′(y))‖ ≤ L‖x− y‖(C5) ‖F′(x∗)−1F′(x)‖ ≤ M.

There exist parameters γi and continuous nondecreasing functions ψi : [0, γi) → R such thati = 0, 1, 2, 3:

(C6) γi+1 ≤ γi ≤ 1L0

and(C7) ψi(t)→ a +∞ or a number greater than 0 as t → γ−1

i . For α ∈ R, consider the functions

qj : [0, γj)→ R j = 0, 1, 2, 3 by

qj(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩M|1− α|, j = 0

Mi+j|1− α|j

∏i=0

ψ1(t) · · ·ψj(t), j = 1, 2, 3

(C8) pj := qj(0) < 1, j = 0, 1, 2, 3,(C9) Ci : Ω → Ω are continuous functions such that for each x ∈ D0, ‖Ci(x)‖ ≤ ψi(‖x− x∗‖) and(C10) U(x∗, r) ⊂ D for some r > 0 to be appointed subsequently.

We are going to introduce some parameters and some functions for the local convergence analysisof the method (2). We define the function g0 on the interval [0, 1

L0) by

g0(t) =1

2(1− L0t)(Lt + 2M|1− α|)

and parameters r0, �A by

r0 =2(1−M|1− α|)

2L0 + L, �A =

22L0 + L

.

Then, since p0 = M|1− α| < 1 by (C8), we have that 0 < r0 < �A, g0(r1) = 1 and for eacht ∈ [0, r1) 0 ≤ g0(t) < 1. Define functions gi, hi on the interval [0, γi) by

gi(t) = (1 +Mψi(t)1− L0t

)gi−1(t)

andhi(t) = gi(t)− 1

for i = 1, 2, 3. We have by (C8) that hi(0) = pj − 1 < 0 and by (C6) and (C7) hi(t)→ a positive numberor +∞. Applying the intermediate value theorem, we know that functions hi have zeros in the interval[0, γi). Denote by ri the smallest such zero. Set

r = min{rj}, j = 0, 1, 2, 3. (3)

Therefore, we can write that0 ≤ r < rA (4)

moreover for each j = 0, 1, 2, 3, t ∈ [0, r)

0 ≤ gj(t) < 1. (5)

236


Now, making use of the conditions (C) and the previous notation, we will show the results oflocal convergence for the method (2).

Theorem 1. Let us assume that (C) conditions hold, if we take the radius r in (C10) that has beendefined previously. Then, the sequence {xn} generated by our method (2) and considering x0 ∈ U(x∗, r) \ {x∗}is well defined, remains in the ball U(x∗, r) for each n ≥ 0 and converges to the solutionx∗. On the other hand,we see that the estimates are true:

‖yn − x∗‖ ≤ g0(‖xn − x∗‖)‖xn − x∗‖ < ‖xn − x∗‖ < r, (6)

‖zn − x∗‖ ≤ g1(‖xn − x∗‖)‖xn − x∗‖ < ‖xn − x∗‖, (7)

‖vn − x∗‖ ≤ g2(‖xn − x∗‖)‖xn − x∗‖ < ‖xn − x∗‖ (8)

and‖xn+1 − x∗‖ ≤ g3(‖xn − x∗‖)‖xn − x∗‖ < ‖xn − x∗‖, (9)

where the “g” functions are defined previously. Furthermore, for

T ∈ [r,2L0

) (10)

the unique solution of equation F(x) = 0 in U(x∗, T) ∩ D is the bound point x∗.

Proof. Using mathematical induction we shall prove estimates (6) and (10). By hypothesis x0 ∈U(x, r) \ {x∗}, the conditions (C1), (C3) and (3), we have that

‖F′(x∗)−1(F′(x0)− F′(x∗))‖ ≤ L0‖x0 − x∗‖ < L0r < 1. (11)

Taking into account the Banach lemma on invertible functions [5,7,39] we can write that F′(x0)−1 ∈

L(S, S) and

‖F′(x0)−1F′(x∗)‖ ≤ 1

1− L0‖x0 − x∗‖ . (12)

consequently, y0 is well defined by the first substep of the method (2) for n = 0. We can set using theconditions (C1) and (C2) that

F(x0) = F(x0)− F(x∗) =∫ 1

0F′(x∗ + θ(x0 − x∗))(x0 − x∗)dθ. (13)

Remark that ‖x∗ + θ(x0 − x∗) − x∗‖ = θ‖x0 − x∗‖ < r, so x∗ + θ(x0 − x∗) ∈ U(x∗, r). Then,using (13) and condition (C5), we have that

‖F′(x∗)−1F(x0)‖ ≤ ‖∫ 1

0F′(x∗)−1F′(x∗ + θ(x0 − x∗))(x0 − x∗)dθ‖ ≤ M‖x0 − x∗‖. (14)

In view of conditions (C2), (C4), (3) and (5) (for j = 0) and (12) and (14), we obtain that

237


‖y0 − x∗‖ = ‖x0 − x∗ − F′(x0)−1F(x0) + (1− α)F′(x0)

−1F(x0)

≤ ‖x0 − x∗ − F′(x0)−1F(x0)‖+ |1− α|‖F(x0)

−1F′(x∗)‖‖F′(x∗)−1F(x0)‖

≤ ‖F(x0)−1F′(x∗)‖‖ ∫ 1

0 F′(x∗)−1(F′(x∗ + θ(x0 − x∗))− F′(x0))(x0 − x∗)dθ

+|1− α|M‖x0 − x∗‖

1− L0‖x0 − x∗‖

≤ L‖x0 − x∗‖2

2(1− L0‖x0 − x∗‖) +|1− α|M‖x0 − x∗‖

1− L0‖x0 − x∗‖= g0(‖x0 − x∗‖)‖x0 − x∗‖ < ‖x0 − x∗‖ < r,

(15)

which evidences (6) for n = 0 and y0 ∈ U(x∗, r). Then, applying (C9) condition, (3) and (5) (for j = 1),(12) and (14) (for y0 = x0) and (15), we achieve that

‖z0 − x∗‖ ≤ g1(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖, (16)

which displays (7) for n = 0 and z0 ∈ U(x∗, r). In the same way, we show estimates (8) and (9) for n = 0and v0, x1 ∈ U(x∗, r). Just substituting x0, y0, z0, v0, x1 by xk, yk, zk, vk, xk+1 in the preceding estimates,we deduct that (6)–(9). Using the estimates ‖xk+1 − x∗‖ ≤ c‖xk − x∗‖ < r, c = g3(‖x0 − x∗‖) ∈ [0, 1),we arrive at lim

k→∞xk = x∗ and xk+1 ∈ U(x∗, r). We have yet to see the uniqueness, let y∗ ∈ U(x∗, T)

be such that F(y∗) = 0. Define B =∫ 1

0 F′(y∗ + θ(x∗ − y∗))dθ. Taking into account the condition (C2),we obtain that

‖F′(x∗)−1(B− F′(x∗))‖ ≤ L0

2‖y∗ − x∗‖ ≤ L0

2T < 1. (17)

Hence, B �= 0. Using the identity 0 = F(y∗)− F(x∗) = B(y∗ − x∗), we can deduct that x∗ =

y∗.

Remark 1.

1. Considering (10) and the next value

‖F′(x∗)−1F′(x)‖ = ‖F′(x∗)−1(I + F′(x)− F′(x∗))‖

≤ ‖F′(x∗)−1(F′(x)− F′(x∗))‖+ 1

≤ L0‖x0 − x∗‖+ 1

we can clearly eliminate the condition (10) and M can be turned into

M(t) = 1 + L0t or what is the same M(t) = M = 2, because t ∈ [0,1L0

).

2. The results that we have seen, can also be applied for F operators that satisfy the autonomous differentialequation [5,7] of the form

F′(x) = P(F(x)),

where P is a known continuous operator. As F′(x∗) = P(F(x∗)) = P(0), we are able to use the previousresults without needing to know the solution x∗. Take for example F(x) = ex − 1. Now, we can takeP(x) = x + 1. However, we do not know the solution.

238


3. In the articles [5,7] was shown that the radius �A has to be the convergence radius for Newton’s methodusing (10) and (11) conditions. If we apply the definition of r1 and the estimates (8), the convergenceradius r of the method (2) it can no be bigger than the convergence radius �A of the second order Newton’smethod. The convergence ball given by Rheinboldt [8] is

�R = 23L1

. (18)

In particular, for L0 < L1 or L < L1 we have that

�R < �A

and�R�A→ 1

3as

L0

L1→ 0.

That is our convergence ball r1 which is maximum three times bigger than Rheinboldt’s. The preciseamount given by Traub in [28] for �R.

4. We should note that family (3) stays the same if we use the conditions of Theorem 1 instead of the strongerconditions given in [15,36]. Concerning, for the error bounds in practice we can use the approximatecomputational order of convergence (ACOC) [36]

ξ =ln ‖xn+2−xn+1‖

‖xn+1−xn‖ln ‖xn+1−xn‖‖xn−xn−1‖

, for each n = 1, 2, . . .

or the computational order of convergence (COC) [40]

ξ∗ =ln ‖xn+2−x∗‖‖xn+1−x∗‖

ln ‖xn+1−x∗‖‖xn−x∗‖

, for each n = 0, 1, 2, . . .

And these order of convergence do not require higher estimates than the first Fréchet derivative usedin [19,23,32,33,41].

Remark 2. Let’s see how we can choose the functions in the case of the method (2). In this case we have that

C1(F(yn)

F(xn)) = C1(xn), C2(

F(yn)

F(xn),

F(zn)

F(yn)) = C2(xn), C3(

F(yn)

F(xn),

F(zn)

F(yn),

F(vn)

F(zn)) = C3(xn)

To begin, the condition (C3) can be eliminated because in this case we have α = 1. Then, if xn �= x∗, thefollowing inequality holds

‖(F′(x∗)(xn − x∗))−1 [F(xn)− F(x∗)− F′(x∗)(xn − x∗)] ‖

≤ ‖xn − x∗‖−1 L0

2‖xn − x∗‖ = L0

2‖xn − x∗‖ < L0

2r < 1.

Hence, we have that

‖F′(xn)−1F(x∗)‖ ≤ 1

‖xn − x∗‖(1− L0

2‖xn − x∗‖)

.

239


Consequently, we get that

‖ F(yn)

F(xn)‖ = ‖F′(xn)−1F′(x∗)‖‖F′(x∗)−1F(yn)‖

≤ M‖yn − x∗‖‖xn − x∗‖(1− L0

2‖x0 − x∗‖)

≤ Mg0(‖xn − x∗‖)1− L0‖xn − x∗‖ .

(19)

Similarly, we obtain

‖F(yn)−1F′(x∗)‖ ≤ 1

‖yn − x∗‖(1− L0

2‖yn − x∗‖)

,

‖ F(zn)

F(yn)‖ ≤

M(1 +Mψ1(‖xn − x∗‖)1− L0‖xn − x∗‖ )

1− L0

2g0(‖xn − x∗‖)‖xn − x∗‖

, (20)

‖F(zn)−1F′(x∗)‖ ≤ 1

‖zn − x∗‖(1− L0

2‖yn − x∗‖)

,

and

‖ F(zn)

F(yn)‖ ≤

M(1 +Mψ2(‖xn − x∗‖)1− L0‖xn − x∗‖ )

1− L0

2g0(‖xn − x∗‖)‖xn − x∗‖

, (21)

Let us choose Ci, i = 1, 2, 3, 4 as in [31]:

C1(a) = 1 + 2a + 4a3 − 3a4 (22)

C2(a, b) = 1 + 2a + b + a2 + 4ab + 3a2b + 4ab2 + 4a3b− 4a2b2 (23)

andC3(a, b, c) = 1 + 2a + b + c + a2 + 4ab + 2ac + 4a2b + a2c + 6ab2 + 8abc− b3 + 2bc. (24)

As these functions, they fulfill the terms imposed in Theorem 1 in [31], So, we have that the order ofconvergence of the method (2) has to reach at least order 16.

Set

a = a(t) =Mg0(t)1− L0t

, (25)

b = b(t) =M(1 +

Mψ1(t)1− L0t

)

1− L0

2t

, (26)

c = c(t) =M(1 +

Mψ2(t)1− L0t

)

1− L0

2t

, (27)

andγi =

1L0

, i = 0, 1, 2, 3.

240


Then it follows from (19)–(24) that functions ψi can be defined by

ψ1(t) = 1 + 2a + 4a3 + 3a4 (28)

ψ2(t) = 1 + 2a + b + a2 + 4ab + 3a2b + 4ab2 + 4a3b + 4a2b2 (29)

andψ3(t) = 1 + 2a + b + c + a2 + 4ab + 2ac + 4a2b + a2c + 6ab2 + 8abc + b3 + 2bc. (30)

3. Dynamical Study of a Special Case of the Family (2)

In this article, the concepts of critical point, fixed point, strange fixed point, attraction basins,parameter planes and convergence planes are going to be assumed. We refer the reader to see [5,7,16,38]to recall the basic dynamical concepts.

In this third section we will study the complex dynamics of a particular case of the method (2),which consists in select:

C1(xn) = F′(yn)−1F′(xn),

C2(xn) = F′(zn)−1F′(xn)

andC3(xn) = F′(yn)

−1F′(xn).

Let be a polynomial of degree two with two roots, that they are not the same. If we apply thisoperator on the previous polynomial and using the Möebius map h(z) = z−A

z−B , we obtain

G(z, α) =z8(1− α + z)8

(−1− z + αz)8 . (31)

The fixed points of this operator are:

• 0• ∞• And 15 more, which are:

– 1 (related to original ∞).– The roots of a 14 degree polynomial.

In Figure 1 the bifurcation diagram of all fixed points, extraneous or not, is presented.

Figure 1. Fixed points’s bifurcation diagram.

241


Now, we are going to compute the critical points, i.e., the roots of

G′(z, α) = − 8(−1+α−z)7z7(−1+α−2z−z2+αz2)(−1−z+αz)9

The free critical points are: cp1(α) = −1 + α, cp2(α) =1−√−(−2+α)α−1+α and cp3(α) =

1+√−(−2+α)α−1+α .

We also have the following results.

Lemma 1.

(a) If α = 0

(i) cp1(α) = cp2(α) = cp3(α) = −1.

(b) If α = 2

(i) cp1(α) = cp2(α) = cp3(α) = 1.

You can easily verify that for every value of α we have to cp2(α) =1

cp3(α)

It is easy to see that there is only one independent critical point. So, we assume that cp2(α) is theonly free critical point without loss of generality. Taking cp2(α), we perform the study of the parameterspace associated with the free critical point. This will allow us to find the some members of the family,and we want to stay with the best members.

We are going to show different planes of parameters. In Figure 2 we show the parameter spacesassociated to critical point cp2(α). Now let us paint a point of cyan if the iteration of the methodstarting in z0 = cp1(α) converges to the fixed point 0 (related to root A) or if it converges to ∞ (alliedto root B). That is, the points relative to the roots of the quadratic polynomial will be painted cyan anda point is painted in yellow if the iteration converges to 1 (related to ∞). Therefore, all convergencewill be painted cyan. On the other hand, convergence to strange fixed points or cycles appears inother colors. As an immediate consequence, all points of the plane that are not cyan are not a goodchoice of α in terms of numerical behavior.

- -

-

-

Figure 2. Parameter space of the free critical point cp2(α).

242


Once we have detected the anomalies, we can go on to describe the dynamic planes. To understandthe colors we have used in these dynamic planes, we have to indicate that if after a maximum of1000 iterations and with a tolerance of 10−6 convergence has not been achieved to the roots, we havepainted in black. Conversely, we colored in magenta the convergence to 0 and colored in cyan theconvergence to ∞. Then, the cyan or magenta regions identify the convergence.

If we focus our attention on the region shown in Figure 2, it is clear that there are family memberswith complicated behaviors. We will also show dynamic planes in Figures 3 and 4, of a family memberwith convergence regions to any of the strange fixed points.

- - ---

Figure 3. Attraction basins associated to α = −10.

---

Figure 4. Attraction basins associated toα = 4.25.

In the following figures, we will show the dynamic planes of family members with convergenceto different attracting n-cycles. For example, in the Figures 5 and 6, we see the dynamic planes to anattracting 2-cycle and in the Figure 7 the dynamic plane of family members with convergence to anattracting 3-cycle that was painted in green in the parameter planes.

243


- - - ---

Figure 5. Attraction basins associated to α = −2.5.

---

Figure 6. Attraction basins associated to α = 11.

- -

---

Figure 7. Attraction basins associated to α = 10− 13i.

Other particular cases are shown in Figures 8 and 9. The basins of attraction for different α valuesin which we see the convergence to the roots of the method can be seen.

244


- -

--

Figure 8. Attraction basins associated to α = 0.5.

- ---

Figure 9. Attraction basins associated to α = −0.5i.

4. Example Applied

Next, we want to show the applicability of the theoretical part previously seen in a real problem.Chemistry is a discipline in which many equations are handled. In this concrete case, let us considerthe quartic equation that can describe the fraction or amount of the nitrogen-hydrogen feed that isturned into ammonia, which is known as fractional conversion and is shown in [42,43].

If the pressure is 250 atm. and the temperature reaches a value of 500 ◦C, the previous equationreduces to: g(x) = x4 − 7.79075x3 + 14.7445x2 + 2.511x− 1.674. We define S as all real line, D as theinterval [0, 1] and ξ = 0. We consider the function F defined on D. If we now take the functions ψi(t)with i = 1, 2, 3 and choosing the value of as α = 1.025, we obtain: L0 = 2.594 . . ., L = 3.282 . . .. It isclear that in this case L0 < L, so we improve the results. Now, we compute M = 1.441 . . .. Additionally,computing the zeros of the functions previously defined, we get: r0 = 0.227 . . ., �A = 0.236 . . .,r1 = 0.082 . . ., r2 = 0.155 . . ., r3 = 0.245 . . ., and as a result of it we get r = r1 = 0.082 . . .. Then wecan guarantee that the method (2) converges for α = 1.025 due to Theorem 1. The applicability of ourfamily of methods is thus proven.

Author Contributions: All authors have contributed equally in writing this article. All authors read and approvedthe final manuscript

Funding: This research was funded by Programa de Apoyo a la investigacin de la fundacin Séneca-Agenciade Ciencia y Tecnología de la Región de Murcia19374/PI/14’ and by the project MTM2014-52016-C2-1-P of theSpanish Ministry of Science and Innovation.

Conflicts of Interest: The authors declare no conflict of interest

245


References

1. Tello, J.I.C.; Orcos, L.; Granados, J.J.R. Virtual forums as a learning method in Industrial EngineeringOrganization. IEEE Latin Am. Trans. 2016, 14, 3023–3028. [CrossRef]

2. LeTendre, G.; McGinnis, E.; Mitra, D.; Montgomery, R.; Pendola, A. The American Journal of Education:Challenges and opportunities in translational science and the grey area of academic. Rev. Esp. Pedag. 2018,76, 413–435. [CrossRef]

3. Argyros, I.K.; González, D. Local convergence for an improved Jarratt–type method in Banach space. Int. J.Interact. Multimed. Artif. Intell. 2015, 3, 20–25. [CrossRef]

4. Argyros, I.K.; George, S. Ball convergence for Steffensen–type fourth-order methods. Int. J. Interact. Multimed.Artif. Intell. 2015, 3, 27–42. [CrossRef]

5. Argyros, I.K.; Magreñán, Á.A. Iterative Methods and Their Dynamics with Applications: A Contemporary Study;CRC-Press: Boca Raton, FL, USA, 2017.

6. Behl, R.; Sarría, Í.; González-Crespo, R.; Magreñán, Á.A. Highly efficient family of iterative methods forsolving nonlinear models. J. Comput. Appl. Math. 2019, 346, 110–132. [CrossRef]

7. Magreñán, Á.A.; Argyros, I.K. A Contemporary Study of Iterative Methods: Convergence, Dynamics andApplications; Elsevier: Amsterdam, The Netherlands, 2018.

8. Rheinboldt, W.C. An adaptive continuation process for solving systems of nonlinear equations. Pol. Acad. Sci.1978, 3, 129–142. [CrossRef]

9. Amat, S.; Busquier, S.; Plaza, S. Dynamics of the King and Jarratt iterations. Aequ. Math. 2005, 69, 212–223.[CrossRef]

10. Amat, S.; Busquier, S.; Plaza, S. Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl.2010, 366, 24–32. [CrossRef]

11. Amat, S.; Hernández, M.A.; Romero, N. A modified Chebyshev’s iterative method with at least sixth orderof convergence. Appl. Math. Comput. 2008, 206, 164–174. [CrossRef]

12. Chicharro, F.; Cordero, A.; Torregrosa, J.R. Drawing dynamical and parameters planes of iterative familiesand methods. Sci. World J. 2013, 2013, 780153. [CrossRef] [PubMed]

13. Gutiérrez, J.M.; Hernández, M.A. Recurrence relations for the super-Halley method. Comput. Math. Appl.1998, 36, 1–8. [CrossRef]

14. Kou, J.; Li, Y. An improvement of the Jarratt method. Appl. Math. Comput. 2007, 189, 1816–1821. [CrossRef]15. Li, D.; Liu, P.; Kou, J. An improvement of the Chebyshev-Halley methods free from second derivative.

Appl. Math. Comput. 2014, 235, 221–225. [CrossRef]16. Magreñán, Á.A. Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput.

2014, 233, 29–38.17. Budzko, D.; Cordero, A.; Torregrosa, J.R. A new family of iterative methods widening areas of convergence.

Appl. Math. Comput. 2015, 252, 405–417. [CrossRef]18. Bruns, D.D.; Bailey, J.E. Nonlinear feedback control for operating a nonisothermal CSTR near an unstable

steady state. Chem. Eng. Sci. 1977, 32, 257–264. [CrossRef]19. Candela, V.; Marquina, A. Recurrence relations for rational cubic methods I: The Halley method. Computing

1990, 44, 169–184. [CrossRef]20. Candela, V.; Marquina, A. Recurrence relations for rational cubic methods II: The Chebyshev method.

Computing 1990, 45, 355–367. [CrossRef]21. Ezquerro, J.A.; Hernández, M.A. New iterations of R-order four with reduced computational cost. BIT Numer.

Math. 2009, 49, 325–342. [CrossRef]22. Ezquerro, J.A.; Hernández, M.A. On the R-order of the Halley method. J. Math. Anal. Appl. 2005, 303,

591–601. [CrossRef]23. Ganesh, M.; Joshi, M.C. Numerical solvability of Hammerstein integral equations of mixed type. IMA J.

Numer. Anal. 1991, 11, 21–31. [CrossRef]24. Hernández, M.A. Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 2001,

41, 433–455. [CrossRef]25. Hernández, M.A.; Salanova, M.A. Sufficient conditions for semilocal convergence of a fourth order multipoint

iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1999, 1, 29–40.

246


26. Jarratt, P. Some fourth order multipoint methods for solving equations. Math. Comput. 1966, 20, 434–437.[CrossRef]

27. Ren, H.; Wu, Q.; Bi, W. New variants of Jarratt method with sixth-order convergence. Numer. Algorithms2009, 52, 585–603. [CrossRef]

28. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall Series in Automatic Computation:Englewood Cliffs, NJ, USA, 1964.

29. Wang, X.; Kou, J.; Gu, C. Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer.Algorithms 2011, 57, 441–456. [CrossRef]

30. Cordero, A.; Torregrosa, J.R.; Vindel, P. Dynamics of a family of Chebyshev-Halley type methods. Appl.Math. Comput. 2013, 219, 8568–8583. [CrossRef]

31. Artidiello, S.; Cordero, A.; Torregrosa, J.R.; Vassileva, M.P. Optimal high order methods for solving nonlinearequations. J. Appl. Math. 2014, 2014, 591638. [CrossRef]

32. Petkovic, M.; Neta, B.; Petkovic, L.; Džunic, J. Multipoint Methods for Solving Nonlinear Equations; Elsevier:Amsterdam, The Netherlands, 2013.

33. Zhao, L.; Wang, X.; Guo, W. New families of eighth-order methods with high efficiency index for solvingnonlinear equations. Wseas Trans. Math. 2012, 11, 283–293.

34. Dzunic, J.; Petkovic, M. A family of Three-Point methods of Ostrowski’s Type for Solving NonlinearEquations. J. Appl. Math. 2012, 2012, 425867. [CrossRef]

35. Chun, C. Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 1990,190, 1432–1437. [CrossRef]

36. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl.Math. Comput. 2007, 190, 686–698. [CrossRef]

37. Ezquerro, J.A.; Hernández, M.A. Recurrence relations for Chebyshev-type methods. Appl. Math. Optim.2000, 41, 227–236. [CrossRef]

38. Magreñán, Á.A. A new tool to study real dynamics: The convergence plane. Appl. Math. Comput. 2014, 248,215–224. [CrossRef]

39. Rall, L.B. Computational Solution of Nonlinear Operator Equations; Robert E. Krieger: New York, NY, USA, 1979.40. Weerakon, S.; Fernando, T.G.I. A variant of Newton’s method with accelerated third-order convergence.

Appl. Math. Lett. 2000, 13, 87–93. [CrossRef]41. Cordero, A.; García-Maimó, J.; Torregrosa, J.R.; Vassileva, M.P.; Vindel, P. Chaos in King’s iterative family.

Appl. Math. Lett. 2013, 26, 842–848. [CrossRef]42. Gopalan, V.B.; Seader, J.D. Application of interval Newton’s method to chemical engineering problems.

Reliab. Comput. 1995, 1, 215–223.43. Shacham, M. An improved memory method for the solution of a nonlinear equation. Chem. Eng. Sci. 1989,

44, 1495–1501. [CrossRef]


247

mathematics

Article

On the Semilocal Convergence of the Multi–PointVariant of Jarratt Method: Unbounded ThirdDerivative Case

Zhang Yong 1,*, Neha Gupta 2, J. P. Jaiswal 2,* and Kalyanasundaram Madhu 3

1 School of Mathematics and Physics, Changzhou University, Changzhou 213164, China2 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462003, India;

[email protected] Department of Mathematics, Saveetha Engineering College, Chennai 602105, India; [email protected]* Correspondence: [email protected] (Z.Y.); [email protected] (J.P.J.)

Received: 10 May 2019; Accepted: 10 June 2019; Published: 13 June 2019

Abstract: In this paper, we study the semilocal convergence of the multi-point variant of Jarrattmethod under two different mild situations. The first one is the assumption that just a second-orderFréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of thenorm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it onthe domain of the nonlinear operator and it also satisfies the local ω-continuity condition in order toprove the convergence, existence-uniqueness followed by a priori error bound. During the study, it isnoted that some norms and functions have to recalculate and its significance can be also seen in thenumerical section.

Keywords: Banach space; semilocal convergence; ω-continuity condition; Jarratt method; error bound

MSC: 65J15; 65H10; 65G99; 47J25

1. Introduction

The problem of finding a solution of the nonlinear equation affects a large area of various fields.For instance, kinetic theory of gases, elasticity, applied mathematics and also engineering dynamicsystems are mathematically modeled by difference or differential equations. Likewise, there arenumerous problems in the field of medical, science, applied mathematics and engineering that canbe reduced in the form of a nonlinear equation. Many of those problems cannot be solved directlythrough any of the methods. For this, we opt for numerical procedure and are able to find at leastan approximate solution of the problem using various iterative methods. In this concern, Newton’smethod [1] is one of the best and most renowned quadratically convergent iterative methods in Banachspaces, which is frequently used by the authors as it is an efficient method and has a smooth execution.Now, consider a nonlinear equation having the form

L(m) = 0, (1)

where L is a nonlinear operator defined as L : B ⊆ ∇1 → ∇2, where B is a non-empty open convexdomain of a Banach space ∇1 with values in a Banach space ∇2 which is usually known as theNewton–Kantorovich method that can be defined as{

m0 given in B,

mn = mn−1 − [L′(mn−1)]−1L(mn−1), n ∈ N,



where L′(mn−1) is the Fréchet derivative of L at mn−1. The results on semilocal convergence have beenoriginally studied by L.V. Kantorovich in [2]. In the early stages, he gave the method of recurrencerelations and afterwards described the method of majorant principle. Subsequently, Rall in [3] andmany researchers have studied the improvements of the results based on recurrence relations. A largenumber of researchers studied iterative methods of various order to solve the nonlinear equationsextensively. The convergence of iterative methods generally relies on two types: semilocal andlocal convergence analysis. In the former type, the convergence of iterative methods depends uponthe information available around the starting point, whereas, in the latter one, it depends on theinformation around the given solution.

In the literature, researchers have developed various higher order schemes in order to get betterefficiency and also discussed their convergence. Various types of convergence analysis using differenttypes of continuity conditions viz. Lipschitz continuity condition has been studied by Wang et al.in [4,5], Singh et al. in [6], and Jaiswal in [7], to name a few. Subsequently, many authors have studiedthe weaker continuity condition than Lipschitz namely Holder by Hernández in [8], Parida andGupta in [9,10], Wang and Kou in [11] are some of them. Usually, there are some nonlinear equationsthat neither satisfy Lipschitz nor Holder continuity conditions; then, we need a generalized formof continuity condition such as ω-continuity, which has been studied by Ezquerro and Hernándezin [12,13], Parida and Gupta in [14,15], Prashanth and Gupta in [16,17], Wang and Kou in [18–20], etc.

The algorithms having higher order of convergence plays an important role where the quickconvergence is required like in the stiff system of equations. Thus, it is quite interesting to studyhigher order methods. In this article, we target our study on the semilocal convergence analysisusing recurrence relations technique on the multi-point variant of Jarratt method when the third orderFréchet derivative becomes unbounded in the given domain.

2. The Method and Some Preliminary Results

Throughout the paper, we use the below mentioned notations:B ≡ non-empty open subset of ∇1; B0 ⊆ B is a non-empty convex subset; ∇1, ∇2 ≡ Banach

spaces, U(m, b) = {n ∈ ∇1 : ‖n−m‖ < b}, U(m, b) = {n ∈ ∇1 : ‖n−m‖ ≤ b}.Here, we consider the multi-point variant of the Jarratt method suggested in [21]

nn = mn +23 (pn −mn),

on = mn − ΥL(mn)℘nL(mn),mn+1 = on −

[ 32 L′(nn)−1ΥL(mn) + ℘n

(I − 3

2 ΥL(mn))]

L(on),(2)

where ΥL(mn) = [6L′(nn)− 2L′(mn)]−1[3L′(nn)+ L′(mn)], ℘n = [L′(mn)]−1, pn = mn−℘nL(mn) andI is the identity operator. In the same article for deriving semilocal convergence results, the researchershave assumed the following hypotheses:

(A1)‖℘0L(m0)‖ ≤ κ,(A2)‖℘0‖ ≤ λ,(A3)‖L′′(m)‖ ≤ P, m ∈ B,(A4)‖L′′′(m)‖ ≤ Q, m ∈ B,(A5)‖L′′′(m)− L′′′(n)‖ ≤ ω(‖m− n‖), ∀ m, n ∈ B,

where ω : R+ → R+, is a continuous and non-decreasing function for m > 0 such that ω(m) ≥ 0 andsatisfying ω(εz) ≤ φ(ε)ω(z), ε ∈ [0, 1] and z ∈ [0,+∞) with φ : [0, 1] → R+, is also continuous andnon-decreasing. One can realize that, if ω(m) = Lm, then this condition is reduced into Lipschitzand when ω(m) = Lmq, q ∈ (0, 1] to the Holder. Furthermore, we found some nonlinear functionswhich are unbounded in a given domain but seem to be bounded on a particular point of the domain.

249


For a motivational example, consider a function h on (−2, 2) . We can verify the above fact byconsidering the following example [22]

h(m) =

{m3ln(m2)− 6m2 − 3m + 8 m ∈ (−2, 0) ∪ (0, 2),

0, m = 0.(3)

Clearly, we can see this fact that h′′′(m) is unbounded in (−2, 2). Hence, for avoiding theunboundedness of the function, we replace the condition (A4) by the milder condition since the givenexample is bounded at m = 1. Thus, here we can assume that the norm of the third order Fréchetderivative is bounded on the initial iterate as:

(B1)‖L′′′(m0)‖ ≤ A, m0 ∈ B0,where m0 be an initial approximation. Moreover, we also assume(B2)‖L′′′(m)− L′′′(n)‖ ≤ ω(‖m− n‖) ∀ m, n ∈ B(m0, ε),where ε > 0. For now, we choose ε = κ

τ0, where τ0 will be defined later and the rationality of this

choice of such ε will be proved. Moreover, some authors have considered partial convergenceconditions. The following nonlinear integral equation of mixed Hammerstein type [23]

m(s) = 1 +∫ 1

0G(s, t)

(12

m(t)52 +

716

m(t)3)

dt, s ∈ [0, 1], (4)

where m ∈ [0, 1], t ∈ [0, 1], G(s, t) is the Green function defined by

G(s, t) =

{(1− s)t t ≤ s,

s(1− t) s ≤ t,

is an example that justified this idea which will be proved later in the numerical application section.In this study, on using recurrence relations, we first discuss the semilocal convergence of theabove-mentioned algorithm by just assuming that the second-order Fréchet derivative is bounded. Inaddition, next, we restrict the domain of the nonlinear operator and consider the bound of the normof the third-order Fréchet derivative on an initial iterate only rather than supposing it on the givendomain of the nonlinear operator.

We start with a nonlinear operator L : B ⊆ ∇1 → ∇2 and let the Hypotheses (A1)–(A3) befulfilled. Consider the following auxiliary scalar functions out of which Δ and Λ function are takenfrom the reference [21] and Γ and Θ have been recalculated:

Γ(θ) = 1 + 12

θ1−θ +

[1 + θ

1− 23 θ

(1 + 1

2θ

1−θ

) ]× θ

2

[1

1−θ +(

1 + 12

θ1−θ

)2]

,(5)

Δ(θ) =1

1− θΓ(θ), (6)

Θ(θ) =

[θ

1− 23 θ

(1 + 1

2θ

1−θ

)+ θ

[1 + θ

1− 23 θ

(1 + 1

2θ

1−θ

) ]+ θ2

2(1−θ)

[1 + θ

1− 23 θ

(1 + 1

2θ

1−θ

) ]+ θ

2

[1 + θ

1− 23 θ

(1 + 1

2θ

1−θ

) ]2

Λ(θ)

]Λ(θ),

(7)

250


where

Λ(θ) =θ

2

[1

1− θ+

(1 +

12

θ

1− θ

)2 ]. (8)

Next, we study some of the properties of the above-stated functions. Let k(θ) = Γ(θ)θ − 1.Since k(0) = −1 < 0 and k( 1

2 ) ≈ 1.379 > 0, then the function k(t) has at least one real root in (0, 12 ).

Suppose γ is the smallest positive root, then clearly γ < 12 . Now, we begin with the following lemmas

that will be used later in the main theorem(s).

Lemma 1. Let the functions Γ, Δ and Θ be given in Equations (5)–(7), respectively, and γ be the smallestpositive real root of Γ(θ)θ − 1. Then,

(a) Γ(θ) and Δ(θ) are increasing and Γ(θ) > 1, Δ(θ) > 1 for θ ∈ (0, γ),(b) for θ ∈ (0, γ), Θ(θ) is an increasing function.

Proof. The proof is straightforward from the expressions of Γ, Δ and Θ given in Relations (5)–(7),respectively.

Define κ0 = κ, λ0 = λ, τ0 = Pλκ and ζ0 = Δ(τ0)Θ(τ0). Furthermore, we designate the followingsequences as:

κn+1 = ζnκn, (9)

λn+1 = Δ(τn)λn, (10)

τn+1 = Pλn+1κn+1 = Δ(τn)ζnτn, (11)

ζn+1 = Δ(τn+1)Θ(τn+1), (12)

where n ≥ 0. Some important properties of the immediate sequences are given by the following lemma.

Lemma 2. If τ0 < γ and Δ(τ0)ζ0 < 1, where γ is the smallest positive root of Γ(θ)θ − 1 = 0, then we have

(a) Δ(τn) > 1 and ζn < 1 for n ≥ 0,(b) the sequences {κn}, {τn} and {ζn} are decreasing,(c) Γ(τn)τn < 1 and Δ(τn)ζn < 1 for n ≥ 0.

Proof. The proof can be done readily using mathematical induction.

Lemma 3. Let the functions Γ, Δ and Θ be given in the Relations (5)–(7), respectively. Assume that α ∈ (0, 1),then Γ(αθ) < Γ(θ), Δ(αθ) < Δ(θ) and Θ(αθ) < α2Θ(θ), for θ ∈ (0, γ).

Proof. For α ∈ (0, 1), θ ∈ (0, γ) and by using the Equations (5)–(7), this lemma can be proved.

3. Recurrence Relations for the Method

Here, we characterized some norms which are already derived in the reference [21] for theMethod (2) and some are recalculated here.

For n = 0, the existence of ℘0 implies the existence of p0, n0 and further, we have

‖p0 −m0‖ ≤ κ0, ‖n0 −m0‖ ≤ 23

κ0, (13)

251


i.e., p0 and n0 ∈ U(m0, ρκ), where ρ = Γ(τ0)1−ζ0

. Let R(m0) = ℘0[L′(n0)− L′(m0)] also; since τ0 < 1, we have

‖R(m0)‖ ≤ 23

τ0 ,

∥∥∥∥∥[

I +32

R(m0)

]−1∥∥∥∥∥ ≤ 1

1− τ0. (14)

Moreover,

‖ΥL(m0)‖ =

∥∥∥∥∥I − 34

[I + 3

2 R(m0)

]−1

R(m0)

∥∥∥∥∥≤ 1 +

∥∥∥∥∥ 34

[I + 3

2 R(m0)

]−1∥∥∥∥∥ ‖R(m0)‖

≤ 1 + 12

τ01−τ0

.

(15)

From the second sub-step of the considered scheme, it is obvious that

‖o0 −m0‖ ≤[

1 +12

τ0

1− τ0

]κ0. (16)

It is similar to obtain

‖o0 − p0‖ ≤[

12

τ0

1− τ0

]κ0. (17)

Using the Banach Lemma, we realize that L′(n0)−1 exists and can be bounded as

‖L′(n0)−1‖ ≤ λ0

1− 23 τ0

. (18)

From Taylor’s formula, we have

L(o0) = L(m0) + L′(m0)(o0 −m0)

+∫ 1

0[L′(m0 + θ(o0 −m0))− L′(m0)]dθ(o0 −m0). (19)

From the above relation, it follows that

‖L(o0)‖ ≤ Λ(τ0)κ

λ. (20)

Though in the considered reference [21] the norm ‖m1 − o0‖ has already been calculated, herewe are recalculating it in a more precise way such that the recalculated norm becomes finer than thegiven in the reference [21] and its significance can be seen in the numerical section. The motivation forrecalculating this norm has been also discussed later. From the last sub-step of the Equation (2),

m1 − o0 = −[

32

L′(n0)−1ΥL(m0) + ℘n

(I − 3

2ΥL(m0)

)]L(o0)

= −[℘0 +

32[L′(n0)

−1 + L′(n0)−1]

]ΥL(m0)L(o0).

On taking the norm, we have

‖m1 − o0‖ ≤ κ02

[1 + τ0

1− 23 τ0

(1 + 1

2τ0

1−τ0

) ]×[

τ01−τ0

+ τ0

(1 + 1

2τ0

1−τ0

)2]

,(21)

252


and thus we obtain

‖m1 −m0‖ ≤ ‖m1 − o0‖+ ‖o0 −m0‖ ≤ Γ(τ0)κ0. (22)

Hence, m1 ∈ U(m0, ρκ). Now, since the assumption ζ0 < 1Δ(τ0)

< 1, notice that τ0 < γ henceΓ(τ0) < Γ(γ) and it can be written as

‖I − ℘0L′(m1)‖ ≤ τ0Λ(τ0) < 1. (23)

Thus, ℘1 = [L′(m1)]−1 exists and, by virtue of Banach lemma, it may be written as

‖℘1‖ ≤ λ0

1− τ0Γ(τ0)= λ1.

Again by Taylor’s expansion along on, we can write

L(mn+1) = L(on) + L′(pn)(mn+1 − on)

+∫ 1

0 [L′(on + θ(mn+1 − on))− L′(pn)]dθ(mn+1 − on),

(24)

and

L′(pn) = L′(mn) +∫ 1

0L′′(mn + θ(pn −mn))dθ(pn −mn). (25)

On using the above relation and, for n = 0, Equation (24) assumes the form

L(m1) = L(o0) + L′(m0)(m1 − o0)

+∫ 1

0L′′(m0 + θ(p0 −m0))dθ(p0 −m0)(m1 − o0)

+∫ 1

0[L′(o0 + θ(m1 − o0))− L′(p0)]dθ(m1 − o0).

Using the last sub-step of the Scheme given in the Equation (2), the above expression can berewritten as

L(m1) =32[L′(n0)− L′(m0)]L′(n0)

−1ΥL(m0)L(o0)

+∫ 1

0L′′(mn + θ(pn −mn))dθ(pn −mn)(mn+1 − on)

+∫ 1

0[L′(on + θ(mn+1 − on))− L′(pn)]dθ(mn+1 − on).

In addition, thus,

‖L(m1)‖ ≤ Θ(τ0)κ

λ. (26)

Hence,‖p1 −m1‖ ≤ Δ(τ0)Θ(τ0)κ0 = κ1.

In addition, because Γ(τ0) > 1 and by triangle inequality, we find

‖p1 −m0‖ ≤ ρκ,

and

‖n1 −m0‖ ≤ ‖m1 −m0‖+∥∥∥∥2

3(p1 −m1)

∥∥∥∥ ≤ (Γ(τ0) + ζ0)κ0 < ρκ,

253


which implies p1, n1 ∈ U(m0, ρκ). Furthermore, we have

P‖℘1‖‖℘1L(m1)‖ ≤ Δ2(τ0)Θ(τ0)τ0 = τ1. (27)

Moreover, we can state the following lemmas.

Lemma 4. Under the hypotheses of Lemma 2, let σ = Δ(τ0)ζ0 and ς = 1Δ(τ0)

, we have

ζi ≤ ςσ3n, (28)

n

∏i=0

ζi ≤ ςn+1σ3n+1−1

2 , (29)

κn ≤ κςnσ3n−1

2 , (30)

n+m

∑i=n

κi ≤ κςnσ3n−1

2

⎛⎝1− ςm+1σ3n(3m+1)

2

1− ςσ3n

⎞⎠ , (31)

where n ≥ 0 and m ≥ 1.

Proof. In order to prove this lemma, first, we need to derive

ζn ≤ ςσ3n.

We will prove it by executing the induction. By Lemma 3 and since τ1 = στ0, hence for n = 1,

ζ1 = Δ(στ0)Θ(στ0) < σ2ζ0 < ςσ31.

Let it be true for n = k, then

ζk ≤ ςσ3k, k ≥ 1.

Now, we will prove it for n = k + 1. Thus,

ζk+1 < Δ(στk)Θ(στk) < ςσ3k+1.

Therefore, ζn ≤ ςσ3nis true for n ≥ 0. Making use of this inequality, we have

k

∏i=0

ζi ≤k

∏i=0

ςσ3i= ςk+1

k

∏i=0

σ3i= ςk+1σ

3k+1−12 , k ≥ 0.

By making use of the above-derived inequality in the Relation (9), we have

κn = ζn−1κn−1 = ζn−1ζn−2κn−2 = · · · = κ0

n−1

∏i=0

ζi ≤ κςnσ3n−1

2 , n ≥ 0.

With the evidence that 0 < ς < 1 and 0 < σ < 1, we can say that κn → 0 as n → ∞. Let us denote

� =k+m

∑i=k

ςiσ3i2 , k ≥ 0, m ≥ 1.

254


The above equation may also be rewritten in the following form

� ≤ ςkσ3k2 + ςσ3k

k+m−1

∑i=k

ςiσ3i2

= ςkσ3k2 + ςσ3k

(�− ςk+mσ

3k+m2

),

and then it becomes

� < ςkσ3k2

⎛⎝1− ςm+1σ3k(3m+1)

2

1− ςσ3k

⎞⎠ .

Moreover,

k+m

∑i=k

κi ≤k+m

∑i=k

κςiσ3i−1

2 ≤ κςkσ3k−1

2

⎛⎝1− ςm+1σ3k(3m+1)

2

1− ςσ3k

⎞⎠ .

Lemma 5. Let the hypotheses of Lemma 2 and the conditions (A1)–(A3) hold; then, the following conditionsare true for all n ≥ 0:

(i)℘n = [L′(mn)]−1exists and ‖℘n‖ ≤ λn,(ii)‖℘nL(mn)‖ ≤ κn,(iii)P‖℘n‖‖℘nL(mn)‖ ≤ τn,(iv)‖pn −mn‖ ≤ κn,(v)‖mn+1 −mn‖ ≤ Γ(τn)κn,(vi)‖mn+1 −m0‖ ≤ ρκ, where ρ = Γ(τ0)

1−ζ0.

(32)

Proof. By using the mathematical induction of Lemma 4, we can prove (i)− (v) for n ≥ 0 . Now,for n ≥ 1, by making use of Relation (31) and the above results, we get

‖mn+1 −m0‖ ≤n

∑i=0‖mi+1 −mi‖ < ρκ.

Lastly, the following lemma can be proved in a similar way of the article by Wang and Kou [22].

Lemma 6. Let ρ = Γ(τ0)1−ζ0

and Δ(τ0)ζ0 < 1 and τ0 < γ, where γ is the smallest positive root of Γ(θ)θ− 1 = 0;then, ρ < 1

τ0.

4. Semilocal Convergence When L′′′ Condition Is Omitted

In the ensuing section, our objective is to prove the convergence of the Algorithm mentioned inthe Equation(2) by assuming the Hypotheses (A1)–(A3) only. Furthermore, we will find a ball withcenter m0 and of radius ρκ in which the solution exists and will be unique as well together with whichwe will define its error bound.

Theorem 1. Suppose L : B ⊆ ∇1 → ∇2 is a continuously second-order Fréchet differentiable on B.Suppose the hypotheses (A1)–(A3) are true and m0 ∈ B. Assume that τ0 = Pλκ and ζ0 = Δ(τ0)Θ(τ0)

satisfy τ0 < γ and Δ(τ0)ζ0 < 1, where γ is the smallest root of Γ(θ)θ − 1 = 0 and Γ, Δ and Θ are definedby Equations (5)–(7), respectively. In addition, suppose U(m0, ρκ) ⊆ B, where ρ = Γ(τ0)

1−ζ0. Then, initiating

255


with m0, the iterative sequence {mn} creating from the Scheme given in the Equation (2) converges to a zerom∗ of L(m) = 0 with mn, m∗ ∈ U(m0, ρκ) and m∗ is an exclusive zero of L(m) = 0 in U(m0, 2

Pλ − ρκ) ∩ B.Furthermore, its error bound is given by

‖mn −m∗‖ ≤ Γ(τ0)κςnσ3n−1

2

(1

1− ςσ3n

), (33)

where σ = Δ(τ0)ζ0 and ς = 1Δ(τ0)

.

Proof. Clearly, the sequence {mn} is well established in U(m0, ρκ). Now,

‖mk+l −mk‖ ≤ ∑k+l−1i=k ‖mi+1 −mi‖

≤ Γ(τ0)κςkσ3k−1

2

(1−ςl σ

3k(3l−1+1)2

1−ςσ3k

),

(34)

which shows that {mk} is a Cauchy sequence. Hence, there exists m∗ satisfying

limk→∞

mk = m∗.

Letting k = 0, l → ∞ in Equation (34), we obtain

‖m∗ −m0‖ ≤ ρκ,

which implies that m∗ ∈ U(m0, ρκ). Next, we will show that m∗ is a zero of L(m) = 0. Because

‖℘0‖ ‖L(mn)‖ ≤ ‖℘n‖ ‖L(mn)‖,

and in the above inequality by tending n → ∞ and using the continuity of L in B, we find thatL(m∗) = 0. Finally, for unicity of m∗ in U(m0, 2

Pλ − ρκ) ∩ B, let m∗∗ be another solution of L(m) inU(m0, 2

Pλ − ρκ) ∩ B. Using Taylor’s theorem, we get

0 = L(m∗∗)− L(m∗) =∫ 1

0L′((1− tθ)m∗ + θm∗∗)dθ(m∗∗ −m∗).

In addition,

‖℘0‖∥∥∥∥∫ 1

0[L′((1− θ)m∗ + θm∗∗)− L′(m0)]dθ

∥∥∥∥≤ Pλ

∫ 1

0[(1− θ)‖m∗ −m0‖+ θ‖m∗∗ −m0‖]dθ

≤ Pλ

2

[ρκ +

2Pλ− ρκ

]= 1,

which implies∫ 1

0 L′((1− θ)m∗ + θm∗∗)dθ is invertible and hence m∗∗ = m∗.

5. Semilocal Convergence When L′′′ Is Bounded on Initial Iterate

In the current section, we establish the existence and uniqueness theorem of the solution based onthe weaker conditions (A1)–(A3), (B1) and (B2). Define the sequences as

κn+1 = ζnκn, (35)

λn+1 = Δ(τn)λn, (36)

256


τn+1 = Pλn+1κn+1 = Δ(τn)ζnτn, (37)

μn+1 = Qλn+1κ2n+1 = Δ(τn)ζ2

nμn, (38)

νn+1 = λn+1κ2n+1ω(κn+1) ≤ Δ(τn)φ(ζn)ζ

2nνn, (39)

ζn+1 = Δ(τn+1)Θ′(τn+1, μn+1, νn+1), (40)

where n ≥ 0 and Q = A + ω(

κτ0

). Here, we assign κ0 = κ, λ0 = λ, τ0 = Pλκ, μ0 = Qλκ2,

ν0 = λκ2ω(κ) and ζ0 = Δ(τ0)Θ′(τ0, μ0, ν0). From Lemma (5), it is known that

‖mn −m0‖ < ρκ <κ

τ0.

Therefore, mn ∈ U(m0, κτ0). Similarly, for t ∈ [0, 1] and n ≥ 1 and using Lemma (6), we get

‖mn + st(pn −mn)−m0‖ ≤ ‖mn −m0‖+ ‖pn −mn‖

≤n−1

∑i=0‖mi+1 −mi‖+ κn

≤ Γ(τ0)n

∑i=0

κi ≤ ρκ <κ

τ0.

Therefore, {mn + st(pn − mn)} ∈ U(m0, κτ0). This shows that the choice for ε = κ

τ0is relevant.

Assume that there exists a root τ0 ∈ (0, γ) of the equation

m =

[A + ω

( κ

m

) ]λκ2.

It is obvious that μ0 = Qλκ2, where Q = A + ω(

κτ0

). Notice that here we don’t define τ0 as the

root of the following equation:

m =

[A + ω

(Γ(m)κ

1− Δ(m)Θ′(m, μ0, ν0)

) ]λκ2.

It would be remembered that, for all m ∈ U(m0, κτ0), we have

‖L′′′(m)‖ = ‖L′′′(m0)‖+ ‖L′′′(m)− L′′′(m0)‖≤ A + ω(‖m−m0‖)≤ A + ω

(κτ0

)= Q.

Here, we include two auxiliary scalar functions taken from the reference [21]

Θ′(θ, η, ξ) =

[56 η + (3θ+η)(6θ+2η)

27−18θ + (2θ+η)(3θ+η)6−4θ

+(2+2θ+η)(3θ+η)θ(12−8θ)(1−θ)

]Λ(θ, η, ξ)

+ 12

θ2

1−θ

[9

6−4θ

(1 + 1

2θ

1−θ

)+ 3θ

4(1−θ)+ 1

2

]Λ(θ, η, ξ)

+ θ2

[9

6−4θ

(1 + 1

2θ

1−θ

)+ 3θ

4(1−θ)+ 1

2

]2

Λ(θ, η, ξ)2,

(41)

257


where

Λ(θ, η, ξ) =18

θ3

(1− θ)2 +112

θη

1− θ+

(D1 +

13

D2

)ξ. (42)

D1 =∫ 1

0

∫ 10 φ(sθ)θ(1− θ)dsdθ and D2 =

∫ 10

∫ 10 φ

( 23 sθ

)θdsdθ.

Using the property of the induction and from the conditions (A1)–(A3), (B1) and (B2), thefollowing relations are true for all n ≥ 0 :

(i)℘n = [L′(mn)]−1exists and ‖℘n‖ ≤ λn,(ii)‖℘nL(mn)‖ ≤ κn,(iii)P‖℘n‖‖℘nL(mn)‖ ≤ τn,(iv)Q‖℘n‖‖℘nL(mn)‖ ≤ μn,(v)‖℘n‖‖℘nL(mn)‖2ω(‖℘nL(mn)‖) ≤ νn,(vi)‖mn+1 −mn‖ ≤ Γ(τn)κn,(vii)‖mn+1 −m0‖ ≤ ρκ, where ρ = Γ(τ0)

1−ζ0.

(43)

The second theorem of this article is based on the weaker assumptions, which is stated as:

Theorem 2. Suppose L : B ⊆ ∇1 → ∇2 is a continuously third-order Fréchet differentiable on a non-emptyopen convex subset B0 ⊆ B. Suppose the hypotheses (A1)–(A3), (B1) and (B2) are true and m0 ∈ B0.Assume that τ0 = Pλκ, μ0 = Qλκ2, ν0 = λκ2ω(κ) and ζ0 = Δ(τ0)Θ′(τ0, μ0, ν0) satisfy τ0 < γ

and Δ(τ0)ζ0 < 1, where γ is the smallest root of Γ(θ)θ − 1 = 0 and Γ, Δ and Θ′ are defined byEquations (5), (6) and (41). In addition, suppose U(m0, ρκ) ⊆ B0, where ρ = Γ(τ0)

1−ζ0. Then, initiating

with m0, the iterative sequence {mn} created from the Scheme given in the Equation (2) converges to a zerom∗ of L(m) = 0 with mn, m∗ ∈ U(m0, ρκ) and m∗ is an exclusive zero of L(m) = 0 in U(m0, 2

Pλ − ρκ) ∩ B.Furthermore, its error bound is given by

‖mn −m∗‖ ≤ Γ(τ0)κςnσ5n−1

4

(1

1− ςσ5n

), (44)

where σ = Δ(τ0)ζ0 and ς = 1Δ(τ0)

.

Proof. Analogous to the proof of Theorem 1.

6. Numerical Example

Example 1. Consider nonlinear integral equation from the reference [23] already mentioned in the introductionis given as

m(s) = 1 +∫ 1

0G(s, t)

(12

m(t)52 +

716

m(t)3)

dt, s ∈ [0, 1], (45)

where m ∈ [0, 1], t ∈ [0, 1] and G is the Green’s function defined by

G(s, t) =

{(1− s)t t ≤ s,

s(1− t) s ≤ t.

258


Proof. Solving Equation (45) is equivalent to find the solution for L(m) = 0, where L : B ⊆ C[0, 1]→C[0, 1] :

[L(m)](s) = m(s)− 1−∫ 1

0G(s, t)

(12

m(t)52 +

716

m(t)3)

dt, s ∈ [0, 1].

The Fréchet derivatives of L are given by

L′(m)n(s) = n(s)−∫ 1

0G(s, t)

(54

m(t)32 +

2116

m(t)2)

n(t)dt, n ∈ B,

L′′(m)no(s) = −∫ 1

0G(s, t)

(158

m(t)12 +

218

m(t))

n(t)o(t)dt, n, o ∈ B.

Using the max-norm and taking into account that a solution m∗ of Equation (45) in C[0, 1] must satisfy

‖m∗‖ − 116‖m∗‖ 5

2 − 7128

‖m∗‖3 − 1 ≤ 0,

i.e., ‖m∗‖ ≤ s1 = 1.18771 and ‖m∗‖ ≥ s2 = 2.54173, where s1 and s2 are the positive roots of the

real equation t− t52

16 − 7128 t3 − 1 = 0. Consequently, if we look for a solution m∗ such that ‖m∗‖ ≤ s1,

we can consider U(0, s) ⊆ C[0, 1], where s ∈ (s1, s2), as a non-empty open convex domain. We choose,for example, s = 2 and therefore B = U(0, 2). If m0 = 1, then

‖℘0‖ = 12887

= λ, ‖℘0L(m0)‖ ≤ 1587

= κ, ‖L′′(m)‖ ≤ 15√

264

+2132

= P.

Thus, τ0 ≈ 0.2505. Hence, τ0Γ(τ0) = 0.4068 < 1 and Δ(τ0)ζ0 = 0.790 < 1 (It is noticeable that, ifwe choose the function Γ(m) from the reference [21], then we get Δ(τ0)ζ0 = 1.280 > 1 which violatesone of the assumed hypotheses considered in Theorem 1 and hence this motivates us to recalculatethe function Γ(m)). In addition, U(m0, ρκ) = U(1, 0.5270) ⊆ U(0, 2) = B. Thus, the conditions ofTheorem 1 of Section 4 are satisfied and the nonlinear Equation (45) has the solution m∗ in the region{u ∈ C[0, 1] : ‖u− 1‖ ≤ 0.5270}, which is unique in {u ∈ C[0, 1] : ‖u− 1‖ < 0.8492} ∩ B. Hence,we can deduce that the existence ball of solution based on our result is superior to that of Wang andKou in [23], but our uniqueness ball is inferior.

Example 2. Now, consider another example discussed in [22] and also mentioned in the introduction, is given by

h(m) =

{m3ln(m2)− 6m2 − 3m + 8, m ∈ (−2, 0) ∪ (0, 2),

0, m = 0.(46)

Proof. Taking U(0, 2) = B. Let m0 = 1 be an initial approximation. The derivatives of h are given by

h′(m) = 3m2ln(m2) + 2m2 − 12m− 3,

h′′(m) = 6mln(m2) + 10m− 12,

h′′′(m) = 6ln(m2) + 22.

Clearly, h′′′ is unbounded in B and does not satisfy the condition (A4) but satisfies assumption(B1), and we have

‖℘0‖ = 113

= λ, ‖℘0h(m0)‖ = 113

= κ, ‖h′′(m)‖ ≤ 12ln(4) + 32 = P.

259


‖h′′′(m0)‖ = 22, ‖h′′′(m)− h′′′(n)‖ ≤ 121− 13

32+12log(4)|m−n|, for all m, n ∈ U

(1, 13

32+12ln(4)

). Here, ω(z) =

121− 13

32+12log(4)z and φ(ε) = 1. Here, τ0 ≈ 0.2878 and since τ0Γ(τ0) = 0.51440 < 1, Δ(τ0)ζ0 = 0.01742 < 1.

Thus, the assumptions of Theorem 2 of Section 5 are satisfied. In addition, thus, the solution lies in theball m ∈ U(1, 0.13867), which is unique in U(1, 0.39592) ∩ B. Table 1 shows the comparison of errorbounds for the considered Algorithm mentioned in the Equation 2 but with two different values offunction Γ(m) (One is given in the reference [21] and the other is recalculated here). This table alsoconfirms that the value of the recalculated function is prominent.

Table 1. Comparison of the error bounds for Method 2.

n With Recalculated Γ(m) With Γ(m) Calculated in [21]

1 0.00085294 0.00191392 1.4091×10−13 2.7117×10−11

3 3.1182×10−61 2.0135×10−48

4 2.9759×10−298 5.9108×10−232

7. Conclusions

In this contribution, we have analyzed the semilocal convergence of a well defined multi-pointvariant of the Jarratt method in Banach spaces. This iterative method can be used to solve variouskinds of nonlinear equations that satisfy the assumed set of hypotheses. The analysis of this methodhas been examined using recurrence relations by relaxing the assumptions in two different approaches.In the first approach, we have softened the classical convergence conditions to the prove convergence,existence and uniqueness results together with a priori error bounds. In another way, we have assumedthe norm of the third order Fréchet derivative on an initial iterate, so that it never gets unbounded onthe given domain and, in addition, it satisfies the local ω-continuity condition as well. Two numericalapplications are mentioned that sustain our theoretical consideration.

Author Contributions: For research All the authors have similar contribution.

Funding: This paper is supported by two project funds: the Natural Science Foundation of the Jiangsu HigherEducation Institutions of China (Grant No: 16KJB110002.) and the National Science Foundation for YoungScientists of China (Grant No: 11701048).

Acknowledgments: The authors are grateful to the reviewers for their significant criticism which made the paper moreelegant and readable.


References

1. Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equation in Several Variables; Academic Press:New York, NY, USA; London, UK, 1970.

2. Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Pergamon Press: Oxford, UK, 1982.3. Rall, L.B. Computational Solution of Nonlinear Operator Equations; Robert E Krieger: New York, NY, USA, 1979.4. Wang, X.; Gu, C.; Kou, J. Semilocal convergence of a multipoint fourth-order Super–Halley method in Banach

spaces. Numer. Algor. 2011, 56, 497–516. [CrossRef]5. Wang, X.; Kou, J.; Gu, C. Semilocal convergence of a sixth-order Jarratt method in Banach spaces.

Numer. Algor. 2011, 57, 441–456. [CrossRef]6. Singh, S.; Gupta, D.K.; Martínez, E.; Hueso, J.L. Semilocal convergence analysis of an iteration of order five

using recurrence relations in Banach spaces. Mediterr. J. Math. 2016, 13, 4219–4235. [CrossRef]7. Jaiswal, J.P. Semilocal convergence of an eighth-order method in Banach spaces and its computational

efficiency. Numer. Algor. 2016, 71, 933–951. [CrossRef]

260


8. Hernández, M.A. Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 2001, 41, 433–445.[CrossRef]

9. Parida, P.K.; Gupta, D.K. Recurrence relations for semilocal convergence of a Newton-like method in Banachspaces. J. Math. Anal. Appl. 2008, 345, 350–361. [CrossRef]

10. Parida, P.K.; Gupta, D.K. Semilocal convergence of a family of third-order methods in Banach spaces underHolder continuous second derivative. Non. Anal. Theo. Meth. Appl. 2008, 69, 4163–4173. [CrossRef]

11. Wang, X.; Kou, J. Convergence for modified Halley-like methods with less computation of inversion. J. Diff.Eqn. Appl. 2013, 19, 1483–1500. [CrossRef]

12. Ezquerro, J.A.; Hernández, M.A. On the R-order of the Halley method. J. Math. Anal. Appl. 2005, 303, 591–601.[CrossRef]

13. Ezquerro, J.A.; Hernández, M.A. A generalization of the Kantorovich type assumptions for Halley’s method.Int. J. Comput. Math. 2007, 84, 1771–1779. [CrossRef]

14. Parida, P.K.; Gupta, D.K. Semilocal convergence of a family of third-order Chebyshev-type methods undera mild differentiability condition. Int. J. Comput. Math. 2010, 87, 3405–3419. [CrossRef]

15. Parida, P.K.; Gupta, D.K.; Parhi, S.K. On Semilocal convergence of a multipoint third order method withR-order (2 + p) under a mild differentiability condition. J. Appl. Math. Inf. 2013, 31, 399–416. [CrossRef]

16. Prashanth, M.; Gupta, D.K. Convergence of a parametric continuation method. Kodai Math. J. 2014, 37, 212–234.[CrossRef]

17. Prashanth, M.; Gupta, D.K. Semilocal convergence for Super-Halley′s method under ω-differentiabilitycondition. Jpn. J. Ind. Appl. Math. 2015, 32, 77–94. [CrossRef]

18. Wang, X.; Kou, J. Semilocal convergence of multi-point improved Super-Halley-type methods without thesecond derivative under generalized weak condition. Numer. Algor. 2016, 71, 567–584. [CrossRef]

19. Wang, X., Kou, J. Semilocal convergence analysis on the modifications for Chebyshev–Halley methods undergeneralized condition. Appl. Math. Comput. 2016, 281, 243–251. [CrossRef]

20. Wang, X.; Kou, J. Semilocal convergence on a family of root-finding multi-point methods in Banach spacesunder relaxed continuity condition. Numer. Algor. 2017, 74, 643–657. [CrossRef]

21. Wang, X.; Kou, J. Semilocal convergence of a modified multi-point Jarratt method in Banach spaces undergeneral continuity condition. Numer. Algor. 2012, 60, 369–390. [CrossRef]

22. Wang, X.; Kou, J. Convergence for a class of improved sixth-order Chebyshev–Halley type method.Appl. Math. Comput. 2016, 273, 513–524. [CrossRef]

23. Wang, X.; Kou, J. Convergence for a family of modified Chebyshev methods under weak condition.Numer. Algor. 2014, 66, 33–48. [CrossRef]


261

mathematics

Article

The Modified Inertial Iterative Algorithm for SolvingSplit Variational Inclusion Problem for Multi-ValuedQuasi Nonexpansive Mappings withSome Applications

Pawicha Phairatchatniyom 1 , Poom Kumam 1,2,* , Yeol Je Cho 3,4,

Wachirapong Jirakitpuwapat 1 and Kanokwan Sitthithakerngkiet 5

1 KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science LaboratoryBuilding, Department of Mathematics, Faculty of Science, King Mongkut’s University of TechnologyThonburi, 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand;[email protected] (P.P.); [email protected] (W.J.)


3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea;[email protected]

4 School of Mathematical Sciences, University of Electronic Science and Technology of China,Chengdu 611731, China

5 Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty ofApplied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Wongsawang,Bangsue, Bangkok 10800, Thailand; [email protected]

* Correspondence: [email protected]; Tel.: +66-24708994

Received: 28 April 2019; Accepted: 28 May 2019; Published: 19 June 2019

Abstract: Based on the very recent work by Shehu and Agbebaku in Comput. Appl. Math. 2017,we introduce an extension of their iterative algorithm by combining it with inertial extrapolationfor solving split inclusion problems and fixed point problems. Under suitable conditions, we provethat the proposed algorithm converges strongly to common elements of the solution set of the splitinclusion problems and fixed point problems.

Keywords: variational inequality problem; split variational inclusion problem; multi-valuedquasi-nonexpasive mappings; Hilbert space

MSC: 47H06; 47H09; 47J05; 47J25

1. Introduction

The split monotone variational inclusion problem (SMVIP) was introduced by Moudafi [1].This problem is as follows:

Find a point x∗ ∈ H1 such that 0 ∈ f (x∗) + B1(x∗) (1)

and such thaty∗ = Ax∗ ∈ H2 solves 0 ∈ g(y∗) + B2(y∗), (2)

where 0 is the zero vector, H1 and H2 are real Hilbert spaces, f and g are given single-valued operatorsdefined on H1 and H2, respectively, B1 and B2 are multi-valued maximal monotone mappings definedon H1 and H2, respectively, and A is a bounded linear operator defined on H1 to H2.



It is well known (see [1]) that

0 ∈ f (x∗) + B1(x∗) ⇐⇒ x∗ = JB1λ (x∗ − λ f (x∗)),

and that0 ∈ g(y∗) + B2(y∗) ⇐⇒ y∗ = JB2

λ (y∗ − λg(y∗)), y∗ = Ax∗,

where JB1λ := (I + λB1)

−1 and JB2λ := (I + λB2)

−1 are the resolvent operators of B1 and B2, respectively,with λ > 0. Note that JB1

λ and JB2λ are nonexpansive and firmly nonexpansive.

Recently, Shehu and Agbebaku [2] proposed an algorithm involving a step-size selectedand proved strong convergence theorem for split inclusion problem and fixed point problemfor multi-valued quasi-nonexpansive mappings. In [1], Moudafi pointed out that the problem(SMVIP) [3–5] includes, as special cases, the split variational inequality problem [6], the split zeroproblem, the split common fixed point problem [7–9] and the split feasibility problem [10,11], whichhave already been studied and used in image processing and recovery [12], sensor networks incomputerized tomography and data compression for models of inverse problems [13].

If f ≡ 0 and g ≡ 0 in the problem (SMVIP), then the problem reduces to the split variationalinclusion problem (SVIP) as follows:

Find a point x∗ ∈ H1 such that 0 ∈ B1(x∗) (3)

and such thaty∗ = Ax∗ ∈ H2 solves 0 ∈ B2(y∗). (4)

Note that the problem (SVIP) is equivalent to the following problem:

Find a point x∗ ∈ H1 such that x∗ = JB1λ (x∗) and y∗ = JB2

λ (y∗), y∗ = Ax∗

for some λ > 0.

We denote the solution set of the problem (SVIP) by Ω, i.e.,

Ω = {x∗ ∈ H1 : 0 ∈ B1(x∗) and 0 ∈ B2(y∗), y∗ = Ax∗}.

Many works have been developed to solve the split variational inclusion problem (SVIP). In 2002,Byrne et al. [7] introduced the iterative method {xn} as follows: For any x0 ∈ H1,

xn+1 = JB1λ (xn + γA∗(JB2

λ − I)Axn) (5)

for each n ≥ 0, where A∗ is the adjoint of the bounded linear operator A, γ ∈ (0, 2/L), L = ‖A∗A‖and λ > 0. They have shown the weak and strong convergence of the above iterative method forsolving the problem (SVIP).

Later, inspired by the above iterative algorithm, many authors have extended the algorithm {xn}generated by (5). In particular, Kazmi and Rizvi [4] proposed an algorithm {xn} for approximating asolution of the problem (SVIP) as follows:{

un = JB1λ (xn + γn A∗(JB2

λ − I)Axn),

xn+1 = αn fn(xn) + (1− αn)Sun(6)

for each n ≥ 0, where {αn} is a sequence in (0, 1), λ > 0, γ ∈ (0, 1/L), L is the spectral radius of theoperator A∗A, f : H1 → H1 is a contraction and S : H1 → H1 is a nonexpansive mapping. In 2015,

263


Sitthithakerngkiet et al. [5] proposed an algorithm {xn} for solving the problem (SVIP) and the fixedpoint problem (FPP) of a countable family of nonexpansive mappings as follows:{

yn = JB1λ (xn + γn A∗(JB2

λ − I)Axn),

xn+1 = αn f (xn) + (1− αnD)Snyn(7)

for each n ≥ 0, where {αn} is a sequence in (0, 1), λ > 0, γ ∈ (0, 1/L), L is the spectral radius of theoperator A∗A, f : H1 → H1 is a contraction, D : H1 → H2 is strongly positive bounded linear operatorand, for each n ≥ 1, Sn : H1 → H1 is a nonexpansive mapping.

In both their works, they obtained some strong convergence results by using their proposediterative methods (for some more results on algorithms, see [14,15]).

Recall that a point x∗ ∈ H1 is called a fixed point of a given multi-valued mapping S : H1 → 2H1

ifx∗ ∈ Sx∗ (8)

and the fixed point problem (FPP) for a multi-valued mapping S : H1 → 2H1 is as follows:

Find a point x∗ ∈ H1 such that x∗ ∈ Sx∗.

The set of fixed points of the multi-valued mapping S is denoted by F(S).

As applications, the fixed point theory for multi-valued mappings was applied to various fields,especially mathematical economics and game theory (see [16–18]).

Recently, motivated by the results of Byrne et al. [7], Kazmi and Rizvi [4] and Sitthithakerngkiet [5],Shehu and Agbebaku [2] introduced the split fixed point inclusion problem (SFPIP) from the problems(SVIP) and (FPP) for a multi-valued quasi-nonexpansive mapping S : H1 → 2H1 as follows:

Find a point x∗ ∈ H1 such that 0 ∈ B1(x∗), x∗ ∈ Sx∗ (9)

and such thaty∗ = Ax∗ ∈ H2 solves 0 ∈ B2(y∗), (10)

where H1 and H2 are real Hilbert spaces, B1 and B2 are multi-valued maximal monotone mappingsdefined on H1 and H2, respectively, and A is a bounded linear operator defined on H1 to H2.

Note that the problem (SFPIP) is equivalent to the following problem: for some λ > 0,

Find a point x∗ ∈ H1 such that x∗ = JB1λ (x∗), x∗ ∈ Sx∗ and Ax∗ = JB2

λ (Ax∗).

The solution set of the problem (SFPIP) is denoted by F(S)⋂

Ω, i.e.,

F(S)⋂

Ω = {x∗ ∈ H1 : 0 ∈ B1(x∗), x∗ ∈ Sx∗ and 0 ∈ B2(Ax∗)}.

Notice that, if S is the identity operator, then the problem (SFPIP) reduces to the problem (SVIP).Moreover, if JB1

λ = JB2λ = A = I, then the problem (SFPIP) reduces to the problem (FPP) for a

multi-valued quasi-nonexpansive mapping.

Furthermore, Shehu and Agbebaku [2] introduced an algorithm {xn} for solving the problem(SFPIP) for a multi-valued quai-nonexpasive mapping S as follows: For any x1 ∈ H1,{

un = JB1λ (xn + γn A∗(JB2

λ − 1)Axn),

xn+1 = αn fn(xn) + βnxn + δn(σwn + (1− σ)un), wn ∈ Sxn,(11)

264


for each n ≥ 1, where {αn}, {βn} and {δn} are the real sequences in (0, 1) such that

αn + βn + δn = 1, σ ∈ (0, 1), γn := τn‖(JB2

λ − I)Axn‖2

‖A∗(JB2λ − I)‖2

,

where 0 < a ≤ τn ≤ b < 1, and { fn(x)} is the uniform convergence sequence for any x in a boundedsubset D of H1, and proved that the sequences {un} and {xn} generated by (11) both converge stronglyto p ∈ F(S) ∩Ω, where p = PF(S)∩Ω f (p).

In optimization theory, the second-order dynamical system, which is called the heavy ball method,is used to accelerate the convergence rate of algorithms. This method is a two-step iterative methodfor minimizing a smooth convex function which was firstly introduced by Polyak [19].

The following is a modified heavy ball method for the improvement of the convergence rate,which was introduced by Nesterov [20]:{

yn = xn + θn(xn − xn−1),

xn+1 = yn − λn∇ f (yn)

for each n ≥ 1, where λn > 0, θn ∈ [0, 1) is an extrapolation factor. Here, the term θn(xn − xn−1) is theinertia (for more recent results on the inertial algorithms, see [21,22]).

The following method is called the inertial proximal point algorithm, which was introduced byAlvarez and Attouch [23]. This method combined the proximal point algorithm [24] with the inertialextrapolation [25,26]: {


xn+1 = (I + λnT)−1(yn)(12)

for each n ≥ 1, where I is identity operator and T is a maximal monotone operator. It was proven that,if a positive sequence λn is non-decreasing, θn ∈ [0, 1) and the following summability condition holds:

∞

∑n=1

θn‖xn − xn−1‖2 < ∞, (13)

then {xn} generated by (12) converges to a zero point of T.

In fact, recently, some authors have pointed out some problems in this summability condition(13) given in [27], that is, to satisfy this summability condition (13) of the sequence {xn}, one needs tocalculate {θn} at each step. Recently, Bot et al. [28] improved this condition, that is, they got rid of thesummability condition (13) and replaced the other conditions.

In this paper, inspired by the results of Shehu and Agbebaku [2], Nesterov [20] and Alvarezand Attouch [23], we proposed a new algorithm by combining the iterative algorithm (11) with theinertial extrapolation for solving the problem (SFPIP) and prove some strong convergence theoremsof the proposed algorithm to show the existence of a solution of the problem (SFPIP). Furthermore, asapplications, we consider our proposed algorithm for solving the variational inequality problem andgive some applications in game theory.

2. Preliminaries

In this section, we recall some definitions and results which will be used in the proof of themain results.

265


Let H1 and H2 be two real Hilbert spaces with the inner product 〈·, ·〉 and the norm ‖ · ‖. LetC be a nonempty closed and convex subset of H1 and D be a nonempty bounded subset of H1. LetA : H1 → H2 be a bounded linear operator and A∗ : H2 → H1 be the adjoint of A.

Let {xn} be a sequence in H, we denote the strong and weak convergence of a sequence {xn} byxn → x and xn ⇀ x, respectively.

Recall that a mapping T : C → C is said to be:

(1) Lipschitz if there exists a positive constant α such that, for all x, y ∈ C,

‖Tx− Ty‖ ≤ α‖x− y‖.

If α ∈ (0, 1) and α = 1, then the mapping T is contractive and nonexpansive, respectively.(2) firmly nonexpansive if

‖Tx− Ty‖2 ≤ 〈Tx− Ty, x− y〉for all x, y ∈ C.

A mapping PC is said to be the metric projection of H1 onto C if, for all point x ∈ H1, there exists aunique nearest point in C, denoted by PCx, such that

‖x− PCx‖ ≤ ‖x− y‖

for all y ∈ C.It is well known that PC is nonexpansive mapping and satisfies

〈x− y, PCx− PCy〉 ≤ ‖PCx− PCy‖2

for all x, y ∈ H1. Moreover, PCx is characterized by the fact PCx ∈ C and

〈x− PCx, y− PCx〉 ≤ 0

for all y ∈ C and x ∈ H1 (see [6,22]).

A multi-valued mapping B1 : H1 → 2H1 is said to be monotone if, for all x, y ∈ H1, u ∈ B1(x) andv ∈ B1(y),

〈x− y, u− v〉 ≥ 0.

A monotone mapping B1 : H1 → 2H1 is said to be maximal if the graph G(B1) of B1 is not properlycontained in the graph of any other monotone mapping. It is known that a monotone mapping B1 ismaximal if and only if, for all (x, u) ∈ H1 × H1,

〈x− y, u− v〉 ≥ 0

for all (y, v) ∈ G(B1) implies that u ∈ B1(x).

Let B1 : H1 → 2H1 be a multi-valued maximal monotone mapping. Then the resolvent mappingJB1λ : H1 → H1 associated with B1 is defined by

JB1λ (x) := (I + λB1)

−1(x)

for all x ∈ H1 and for some λ > 0, where I is the identity operator on H1. It is well known that, for anyλ > 0, the resolvent operator JB1

λ is single-valued firmly nonexpansive (see [2,5,6,14]).

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Definition 1. Suppose that { fn(x)} is a sequence of functions defined on a bounded set D. Then fn(x)converges uniformly to the function f (x) on D if, for all x ∈ D,

fn(x)→ f (x) as n → ∞.

Let fn : D → H1 be a uniformly convergent sequence of contraction mappings on D, i.e., thereexists μn ∈ (0, 1) such that

fn(x)− fn(y)‖ ≤ μn‖x− y‖for all x, y ∈ D.

Let CB(H1) denote the family of nonempty closed and bounded subsets of H1. The Hausdorffmetric on CB(H1) is defined by

H(x, y) = max

{supx∈A

infy∈B‖x− y‖, sup

y∈Binfx∈A

‖x− y‖}

for all A, B ∈ CB(H1) (see [18]).

Definition 2. [2] Let S : H1 → CB(H1) be a multi-valued mapping. Assume that p ∈ H1 is a fixed point ofS, that is, p ∈ Sp. The mapping S is said to be:

(1) nonexpansive if, for all x, y ∈ H1,H(Sx, Sy) ≤ ‖x− y‖.

(2) quasi-nonexpansive if F(S) �= ∅ and, for all x ∈ H1 and p ∈ F(S),

H(Sx, Sp) ≤ ‖x− p‖

Definition 3. [2] A single-valued mapping S : H → H is said to be demiclosed at the origin if, for anysequence {xn} ⊂ H with xn ⇀ x and Sxn → 0, we have Sx = 0.

Definition 4. [2] A multi-valued mapping S : H1 → CB(H1) is said to be demiclosed at the origin if, forany sequence {xn} ⊂ H with xn ⇀ x and d(xn, Sxn)→ 0, we have x ∈ Sx.

Lemma 1. [29,30] Let H be a Hilbert space. Then, for any x, y, z ∈ X and α, β, γ ∈ [0, 1] with α + β + γ = 1,we have

‖αx + βy + γz‖2 = α‖x‖2 + β‖y‖2 + γ‖z‖2 − αβ‖x− y‖2 − αγ‖x− z‖2 − βγ‖y− z‖2.

Lemma 2. [2,31] Let H be a real Hilbert space. Then the following results hold:

(1) ‖x− y‖2 = ‖x‖2 − 2〈x, y〉+ ‖y‖2.(2) ‖x + y‖2 = ‖x‖2 + 2〈x, y〉+ ‖y‖2.(3) ‖x + y‖2 ≤ ‖x‖2 + 2〈y, x + y〉 for all x, y ∈ H.

Lemma 3. [2,32,33] Let {an}, {cn} ⊂ R+, {σn} ⊂ (0, 1) and {bn} ⊂ R be sequences such that

an+1 ≤ (1− σn)an + bn + cn for all n ≥ 0.

Assume ∑∞n=0 |cn| < ∞. Then the following results hold:

(1) If bn ≤ βσn for some β ≥ 0, then {an} is a bounded sequence.

267


(2) If we have∞

∑n=0

σn = ∞ and lim supn→∞

bn

σn≤ 0,

then limn→∞ an = 0.

Lemma 4. [32,33] Let {sn} be a sequence of non-negative real numbers such that

sn+1 ≤ (1− λn)sn + λntn + rn

for each n ≥ 1, where

(a) {λn} ⊂ [0, 1] and ∑∞n=1 λn = ∞;

(b) lim sup tn ≤ 0;(c) rn ≥ 0 and ∑∞

n=1 rn < ∞.

Then sn → 0 as n → ∞.

3. The Main Results

In this section, we prove some strong convergence theorems of the proposed algorithm for solvingthe problem (SFPIP).

Theorem 1. Let H1, H2 be two real Hilbert spaces, A : H1 → H2 be bounded operator with adjoint operator A∗

and B1 : H1 → 2H1 , B2 : H2 → 2H2 be maximal monotone mappings. Let S : H1 → CB(H1) be a multi-valuedquasi-nonexpansive mapping and S be demiclosed at the origin. Let { fn} be a sequence of μn-contractionsfn : H1 → H1 with 0 < μ∗ ≤ μn ≤ μ∗ < 1 and { fn(x)} be uniformly convergent for any x in a boundedsubset D of H1. Suppose that F(S) ∩Ω �= ∅. For any x0, x1 ∈ H1, let the sequences {yn}, {un}, {zn} and{xn} be generated by ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩


un = JB1λ (yn + γn A∗(JB2

λ − I)Ayn),

zn = ξvn + (1− ξ)un, vn ∈ Sxn,

xn+1 = αn fn(xn) + βnxn + δnzn

(14)

for each n ≥ 1, where ξ ∈ (0, 1), γn := τn‖(JB2

λ −I)Ayn‖2

‖A∗(JB2λ −I)Ayn‖2

with 0 < τ∗ ≤ τn ≤ τ∗ < 1, {θn} ⊂ [0, ω) for

some ω > 0 and {αn}, {βn}, {δn} ∈ (0, 1) with αn + βn + δn = 1 satisfying the following conditions:

(C1) limn→∞ αn = 0;(C2) ∑∞

n=1 αn = ∞;(C3) 0 < ε1 ≤ βn and 0 < ε2 ≤ δn;(C4) limn→∞

θnαn‖xn − xn−1‖ = 0.

Then {xn} generated by (14) converges strongly to p ∈ F(S) ∩Ω, where p = PF(S)∩Ω f (p).

Proof. First, we show that {xn} is bounded. Let p = PF(S)∩Ω f (p). Then p ∈ F(S)∩Ω and so JB1λ p = p

and JB2λ Ap = Ap. By the triangle inequality, we get

‖yn − p‖ = ‖xn + θn(xn − xn−1)− p‖≤ ‖xn − p‖+ θn‖xn − xn−1‖. (15)

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By the Cauchy-Schwarz inequality and Lemma 2 (1) and (2), we get

‖yn − p‖2 = ‖xn + θn(xn − xn−1)− p‖2

= ‖xn − p‖2 + θ2n‖xn − xn−1‖2 + 2θn〈xn − p, xn − xn−1〉

≤ ‖xn − p‖2 + θ2n‖xn − xn−1‖2 + 2θn‖xn − xn−1‖‖xn − p‖. (16)

By using (15) and the fact that S is quasi-nonexpansive S, we get

‖zn − p‖ = ‖ξvn + (1− ξ)un − p‖= ‖ξ(vn − p) + (1− ξ)(un − p)‖≤ ξ‖vn − p‖+ (1− ξ)‖un − p‖≤ ξd(vn, Sp) + (1− ξ)‖yn − p‖≤ ξH(Sxn, Sp) + (1− ξ)[‖xn − p‖+ θn‖xn − xn−1‖]≤ ξ‖xn − p‖+ (1− ξ)‖xn − p‖+ (1− ξ)θn‖xn − xn−1‖≤ ‖xn − p‖+ θn‖xn − xn−1‖, (17)

which implies that

‖zn − p‖2 ≤ (‖xn − p‖+ θn‖xn − xn−1‖)2

= ‖xn − p‖2 + 2θn‖xn − xn−1‖‖xn − p‖+ θ2n‖xn − xn−1‖2. (18)

Since JB1λ is nonexpansive, by Lemma 2 (2), we get

‖un − p‖2 = ‖JBIλ (yn + γn A∗(JB2

λ − I)Ayn)− p‖2

= ‖JB1λ (yn + γn A∗(JB2

λ − I)Ayn)− JB1λ p‖2

≤ ‖yn + γn A∗(JB2λ − I)Ayn − p‖2

= ‖yn − p‖2 + γ2n‖A∗(JB2

λ − I)Ayn‖2 + 2γn〈yn − p, A∗(JB2λ − I)Ayn〉. (19)

Again, by Lemma 2 (2), we get

〈yn − p, A∗(JB2λ − I)Ayn〉

= 〈A(yn − p), (JB2λ − I)Ayn〉

= 〈JB2λ Ayn − Ap− (JB2

λ − I)Ayn, (JB2λ − I)Ayn〉

= 〈JB2λ Ayn − Ap, (JB2

λ − I)Ayn〉 − 〈(JB2λ − I)Ayn, (JB2

λ − I)Ayn〉= 〈JB2

λ Ayn − Ap, (JB2λ − I)Ayn〉 − ‖(JB2

λ − I)Ayn‖2

=12(‖JB2

λ Ayn − Ap‖2 + ‖(JB2λ − I)Ayn‖2

− ‖JB2λ Ayn − Ap− (JB2

λ − I)Ayn‖2)− ‖(JB2λ − I)Ayn‖2

=12(‖JB2

λ Ayn − Ap‖2 + ‖(JB2λ − I)Ayn‖2 − ‖JB2

λ Ayn − Ap− JB2λ Ayn + Ayn‖2)

− ‖(JB2λ − I)Ayn‖2

=12(‖JB2

λ Ayn − Ap‖2 + ‖(JB2λ − I)Ayn‖2 − ‖Ayn − Ap‖2)− ‖(JB2

λ − I)Ayn‖2

=12(‖JB2

λ Ayn − Ap‖2 − ‖Ayn − Ap‖2 − ‖(JB2λ − I)Ayn‖2)

≤ 12(‖Ayn − Ap‖2 − ‖Ayn − Ap‖2 − ‖(JB2

λ − I)Ayn‖2)= −1

2‖(JB2

λ − I)Ayn‖2. (20)

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Using (20) into (19), we get

‖un − p‖2 ≤ ‖yn − p‖2 + γ2n‖A∗(JB2

λ − I)Ayn‖2 − γn‖(JB2λ − I)Ayn‖2

= ‖yn − p‖2 − γn(‖(JB2

λ − I)Ayn‖2 − γn‖A∗(JB2λ − I)Ayn‖2). (21)

By the definition of γn, (21) can then be written as follows:

‖un − p‖2 ≤ ‖yn − p‖2 − γn(1− τn)‖(JB2λ − I)Ayn‖2 ≤ ‖yn − p‖2.

Thus we have‖un − p‖ ≤ ‖yn − p‖. (22)

Using the condition (C3) and (17), we get

‖xn+1 − p‖ = ‖αn fn(xn) + βnxn + δnzn − p‖= ‖αn( fn(xn)− fn(p)) + αn( fn(p)− p) + βn(xn − p) + δn(zn − p)‖≤ αn‖ fn(xn)− fn(p)‖+ αn‖ fn(p)− p‖+ βn‖xn − p‖+ δn‖zn − p‖≤ αnμn‖xn − p‖+ αn‖ fn(p)− p‖+ βn‖xn − p‖+ δn(‖xn − p‖+ (1− ξ)θn‖xn − xn−1‖)

≤ (αnμ∗ + (βn + δn))‖xn − p‖+ (1− ξ)δnθn‖xn − xn−1‖+ αn‖ fn(p)− p‖= (1− αn(1− μ∗)‖xn − p‖+ (1− ξ)δnαn

θn

αn‖xn − xn−1‖+ αn‖ fn(p)− p‖.

Since { fn} is the uniform convergence on D, there exists a constant M > 0 such that

‖ fn(p)− p‖ ≤ M

for each n ≥ 1. So we can choose β :=M

1− μ∗ and set

an := ‖xn − p‖, bn := αn‖ fn(p)− p‖,

cn := (1− ξ)δnαnθn

αn‖xn − xn−1‖, σn := αn(1− μ∗).

By Lemma 3 (1) and our assumptions, it follows that {xn} is bounded. Moreover, {un} and {yn} arealso bounded.

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Now, by Lemma 2, we get

‖xn+1 − p‖2

= ‖αn( fn(xn)− fn(p)) + αn( fn(p)− p) + βn(xn − p) + δn(zn − p)‖2

≤ ‖αn( fn(xn)− fn(p)) + βn(xn − p) + δn(zn − p)‖2 + 2αn〈 fn(p)− p, xn+1 − p〉= ‖βn(xn − p) + δn(zn − p)‖2 + α2

n‖ fn(xn)− fn(p)‖2

+ 2αn〈 fn(xn)− fn(p), βn(xn − p) + δn(zn − p)〉+ 2αn〈 fn(p)− p, xn+1 − p〉≤ β2

n‖xn − p‖2 + δ2n‖zn − p‖2 + 2βnδn〈xn − p, zn − p〉+ α2

nμ2n‖xn − p‖2

+ 2αn〈 fn(p)− p, xn+1 − p〉+ 2αn‖ fn(xn)− fn(p)‖‖βn(xn − p) + δn(zn − p)‖≤ β2

n‖xn − p‖2 + δ2n‖zn − p‖2 + βnδn

(‖xn − p‖2 + ‖zn − p‖2 − ‖xn − zn‖2

)+ α2

nμ∗2‖xn − p‖2 + 2αnμn‖xn − p‖ (βn‖xn − p‖+ δn‖zn − p‖)+ 2αn〈 fn(p)− p, xn+1 − p〉

≤ βn(βn + δn)‖xn − p‖2 + δn(βn + δn)‖zn − p‖2 − βnδn‖xn − zn‖2 + α2nμ∗2‖xn − p‖2

+ 2μ∗αn(βn + δn)‖xn − p‖2 + 2μ∗αn(1− ξ)δnθn‖xn − xn−1‖‖xn − p‖+ 2αn〈 fn(p)− p, xn+1 − p〉

≤ βn(βn + δn)‖xn − p‖2 + δn(βn + δn)(‖xn − p‖2 + θ2

n‖xn − xn−1‖2

+ 2θn‖xn − xn−1‖‖xn − p‖)− βnδn‖xn − zn‖2 + α2nμ∗2‖xn − p‖2

+ 2μ∗αn(βn + δn)‖xn − p‖2 + 2μ∗αn(1− ξ)δnθn‖xn − xn−1‖‖xn − p‖+ 2αn〈 fn(p)− p, xn+1 − p〉

=((1− αn)

2 + α2nμ∗2 + 2μ∗αn(1− αn)

)‖xn − p‖2 − βnδn‖xn − zn‖2

+ 2(1− αn(1− μ∗(1− ξ))

)δnθn‖xn − xn−1‖‖xn − p‖+ (1− αn)δnθ2

n‖xn − xn−1‖2

+ 2αn〈 fn(p)− p, xn+1 − p〉. (23)

Now, we consider two steps for the proof as follows:

Case 1. Suppose that there exists n0 ∈ N such that {‖xn − p‖}∞n=n0

is non-increasing and then{‖xn − p‖} converges. By Lemma 1, we get

‖xn+1 − p‖2 = ‖αn fn(xn) + βnxn + δnzn − p‖2

= αn‖ fn(xn)− p‖2 + βn‖xn − p‖2 + δn‖zn − p‖2 − αnβn‖ fn(xn)− xn‖2

− αnγn‖ fn(xn)− zn‖2 − βnγn‖xn − zn‖2

≤ αn‖ fn(xn)− p‖2 + βn‖xn − p‖2 + δn‖zn − p‖2

≤ αn‖ fn(xn)− p‖2 + βn‖xn − p‖2 + δn(ξ‖xn − p‖2 + (1− ξ)‖un − p‖2)

≤ αn‖ fn(xn)− p‖2 + (βn + ξδn)‖xn − p‖2 + (1− ξ)δn‖un − p‖2,

which implies that

−‖un − p‖2 ≤ 1(1− ξ)δn

(αn‖ fn(xn)− p‖2 + (βn + ξδn)‖xn − p‖2 − ‖xn+1 − p‖2). (24)

271


Applying (16) and (24) to (21), we get

γn(‖(JB2λ − I)Ayn‖2 − γn‖A∗(JB2

λ − I)Ayn‖2)

≤ ‖yn − p‖2 − ‖un − p‖2

≤ ‖xn − p‖2 + 2θn‖xn−1 − p‖‖xn − p‖+ θ2n‖xn − xn−1‖2

+1

(1− ξ)δn(αn‖ fn(xn)− p‖2 + (βn + ξδn)‖xn − p‖2 − ‖xn+1 − p‖2)

=βn + δn

(1− ξ)δn‖xn − p‖2 +

αn

(1− ξ)δn‖ fn(xn)− p‖2 − 1

(1− ξ)δn‖xn+1 − p‖2

+ θn‖xn − xn−1‖ (2‖xn − p‖+ θn‖xn − xn−1‖)≤ 1

(1− ξ)ε2(‖xn − p‖2 − ‖xn+1 − p‖2) +

αn

(1− ξ)ε2

(‖ fn(xn)− p‖2 − ‖xn − p‖2

+θn

αn‖xn − xn−1‖

(2‖xn − p‖+ αn

θn


)).

Since {‖xn − p‖} is convergent, we have ‖xn − p‖ − ‖xn+1 − p‖ → 0 as n → ∞. By the conditions(C2) and (C4), we get

γn(‖(JB2λ − I)Ayn‖2 − γn‖A∗(JB2

λ − I)Ayn‖2)→ 0 as n → ∞.

From the definition of γn, we get

τn(1− τn)‖(JB2λ − I)Ayn‖4

‖A∗(JB2λ − I)Ayn‖2

→ 0 as n → ∞

or‖(JB2

λ − I)Ayn‖2

‖A∗(JB2λ − I)Ayn‖

→ 0 as n → ∞.

Since

‖A∗(JB2λ − I)Ayn‖ ≤ ‖A∗‖‖(JB2

λ − I)Ayn‖ = ‖A‖‖(JB2λ − I)Ayn‖,

it is easy to see that

‖(JB2λ − I)Ayn‖ ≤ ‖A‖ ‖(JB2

λ − I)Ayn‖2

‖A∗(JB2λ − I)Ayn‖

.

Consequently, we get

‖(JB2λ − I)Ayn‖ → 0 as n → ∞ (25)

and also‖A∗(JB2

λ − I)Ayn‖ → 0 as n → ∞. (26)

272


Similarly, from (23) and our assumptions, we get

‖xn − zn‖2

=1

βnδn

{‖xn − p‖2 − ‖xn+1 − p‖2 + (1− αn)δnθ2n‖xn − xn−1‖2

+ 2(1− αn(1− μ∗(1− ξ))

)δnθn‖xn − xn−1‖‖xn − p‖

+ αn[(

αn(1 + μ∗2)− 2(1− μ∗(1− αn)))‖xn − p‖2 + 2〈 fn(p)− p, xn+1 − p〉]}

≤ 1ε1ε2

{‖xn − p‖2 − ‖xn+1 − p‖2 +θn


[δn(1− αn)α

2n

θn


+ 2δn(1− αn(1− μ∗(1− ξ))

)θn‖xn − p‖]+ αn

[2〈 fn(p)− p, xn+1 − p〉

+(αn(1 + μ∗2)− 2(1− μ∗(1− αn))

)‖xn − p‖2]}→ 0 as n → ∞.

Therefore, we have

‖xn − zn‖ → 0 as n → ∞. (27)

By the condition (C2) and (27), we get

‖xn+1 − xn‖ = ‖αn fn(xn) + βnxn + δnzn − xn‖≤ αn‖ fn(xn)− xn‖+ δn‖xn − zn‖ → 0 as n → ∞.

Thus we have

‖xn+1 − zn‖ ≤ ‖xn+1 − xn‖+ ‖xn − zn‖ → 0 as n → ∞.

Since JB1λ is firmly nonexpansive, we have

‖un − p‖2

= ‖JB1λ (yn + γn A∗(JB2

λ − I)Ayn)− JB1λ p‖2

≤ 〈un − p, yn + γn A∗(JB2λ − I)Ayn − p〉

=12(‖un − p‖2 + ‖yn + γn A∗(JB2

λ − I)Ayn − p‖2 − ‖un − yn − γn A∗(JB2λ − I)Ayn‖2)

=12(‖un − p‖2 + ‖yn − p‖2 + γ2

n‖A∗(JB2λ − I)Ayn‖2 + 2〈yn − p, γn A∗(JB2

λ − I)Ayn〉− ‖un − yn‖2 − γ2

n‖A∗(JB2λ − I)Ayn‖2 + 2〈un − yn, γn A∗(JB2

λ − I)Ayn〉)

≤ 12(‖yn − p‖2 + ‖yn − p‖2 − ‖un − yn‖2 + 2〈un − p, γn A∗(JB2

λ − I)Ayn〉)

≤ 12(2‖yn − p‖2 − ‖un − yn‖2 + 2γn‖un − p‖‖A∗(JB2

λ − I)Ayn‖)

≤ ‖yn − p‖2 − 12‖un − yn‖2 + γn‖un − p‖‖A∗(JB2

λ − I)Ayn‖

or‖un − yn‖2 ≤ 2

(‖yn − p‖2 − ‖un − p‖2 + γn‖un − p‖‖A∗(JB2λ − 1)Ayn‖

). (28)

273


From (28), (16), (24) and (26) and our assumptions, it follows that

‖un − yn‖2 ≤ 2[‖xn − p‖2 + 2θn‖xn − xn−1‖‖xn − p‖+ θ2

n‖xn − xn−1‖2

+1

(1− ξ)δn

(αn‖ fn(xn)− p‖2 + (βn + ξδn)‖xn − p‖2 − ‖xn+1 − p‖2)

+ γn‖un − p‖‖A∗(JB2λ − 1)Ayn‖

]= 2

[ 1(1− ξ)ε2

(‖xn − p‖2 − ‖xn+1 − p‖2)+ γn‖un − p‖‖A∗(JB2λ − 1)Ayn‖

+αn

(1− ξ)ε2

(‖ fn(xn)− p‖2 − ‖xn − p‖2

+θn


(2‖xn − p‖+ αn

θn


))]→ 0 as n → ∞,

that is, we have‖un − yn‖ → 0 as n → ∞. (29)

From yn := xn + θn(xn − xn−1), we get

‖yn − xn‖ = ‖xn + θn(xn − xn−1)− xn‖ = αnθn

αn‖xn − xn−1‖,

which, with the condition (C4), implies that

‖yn − xn‖ → 0 as n → ∞. (30)

In addition, using (27), (29) and (30), we obtain

‖zn − un‖ ≤ ‖un − yn‖+ ‖yn − zn‖≤ ‖un − yn‖+ ‖yn − xn‖+ ‖xn − zn‖ → 0 as n → ∞.

From zn := ξvn + (1− ξ)un, we get

‖vn − un‖ = 1ξ‖zn − un‖ → 0 as n → ∞. (31)

Thus, by (29)–(31), we also get

‖xn − vn‖ ≤ ‖xn − un‖+ ‖un − vn‖≤ ‖xn − yn‖+ ‖yn − un‖+ ‖un − vn‖ → 0 as n → ∞.

Therefore, we haved(xn, Sxn) ≤ ‖xn − vn‖ → 0 as n → ∞. (32)

Since {xn} is bounded, there exists a subsequence {xnk} of {xn} such that xnk ⇀ x∗ ∈ H1 and,consequently, {unk} and {ynk} converge weakly to the point x∗.

From (32), Lemma 4 and the demiclosedness principle for a multi-valued mapping S at the origin,we get x∗ ∈ Sx∗, which implies that

x∗ ∈ F(S).

Next, we show that x∗ ∈ Ω. Let (v, z) ∈ G(B1), that is, z ∈ B1(v). On the other hand,unk = JB1

λ (ynk + γnk A∗(JB2λ − I)Aynk ) can be written as

ynk + γnk A∗(JB1λ − I)Aynk ∈ unk + λB1(unk ),

274


or, equivalently,(ynk − unK ) + γnk A∗(JB1

λ − I)Aynk

λ∈ B1(unk ).

Since B1 is maximal monotone, we get

⟨v− unk , z− (ynk − unk ) + γnk A∗(JB2

λ − I)Aynk

λ

⟩≥ 0.

Therefore, we have

〈v− unk , z〉 ≥⟨

v− unk ,(ynk − unk ) + γnk A∗(JB2

λ − I)Aynk

λ

⟩=⟨

v− unk ,ynk − unk

λ

⟩+⟨

v− unk ,γnk A∗(JB2

λ − I)Aynk

λ

⟩. (33)

Since unk ⇀ x∗, we havelimk→∞

〈v− unk , z〉 = 〈v− x∗, z〉.

By (26) and (29), it follows that (33) becomes 〈v− x∗, z〉 ≥ 0, which implies that

0 ∈ B1(x∗).

Moreover, from (29), we know that {Aynk} converges weakly to Ax∗ and, by (25), the fact that JB2λ

is nonexpansive and the demiclosedness principle for a multi-valued mapping, we have

0 ∈ B2(Ax∗),

which implies that x∗ ∈ Ω. Thus x∗ ∈ F(S) ∩Ω. Since { fn(x)} is uniformly convergent on D, we get

lim supn→∞

〈 fn(p)− p, xn+1 − p〉 = lim supj→∞

〈 fnj(p)− p, xnj+1 − p〉

= 〈 f (p)− p, x∗ − p〉 ≤ 0.

From (23), we get

‖xn+1 − p‖2 ≤ (1− 2αn(1− μ∗(1− αn)) + α2

n(1 + μ∗2))‖xn − p‖2 − βnδn‖xn − zn‖2

+ 2(1− αn(1− μ∗(1− ξ))

)δnθn‖xn − xn−1‖‖xn − p‖

+ (1− αn)δnθ2n‖xn − xn−1‖2 + 2αn〈 fn(p)− p, xn+1 − p〉

≤ (1− 2αn(1− μ∗)

)‖xn − p‖2 + 2αn(1− μ∗) 〈 fn(p)− p, xn+1 − p〉1− μ∗

+ αn[δn

θn


(2(1− αn(1− μ∗(1− ξ))

)‖xn − p‖

+((1− αn)αn

θn


)+ αn(1 + μ∗2)‖xn − p‖2].

By Lemma 4, we obtainlim

n→∞xn = p.

Case 2. Suppose that {‖xn − p‖}∞n=n0

is not a monotonically decreasing sequence for some n0 largeenough. Set Γn = ‖xn − p‖2 and let τ : B→ N be a mapping defined by

τ(n) := max{k ∈ N : k ≤ n, Γk ≤ Γk+1}

275


for all n ≥ n0. Obviously, τ is a non-decreasing sequence. Thus we have

0 ≤ Γτ(n) ≤ Γτ(n)+1

for all n ≥ n0. That is, ‖xτ(n) − p‖ ≤ ‖xτ(n)+1 − p‖ for all n ≥ n0. Thus limn→∞ ‖xτ(n) − p‖ exists.As in Case 1, we can show that

limn→∞

‖(JB2λ − I)Ayτ(n)‖ = 0, lim

n→∞‖A∗(JB2

λ − I)Ayτ(n)‖ = 0, (34)

limn→∞

‖xτ(n)+1 − xτ(n)‖ = 0, limn→∞

‖uτ(n) − xτ(n)‖ = 0, (35)

limn→∞

‖vτ(n) − uτ(n)‖ = 0, limn→∞

‖xτ(n) − vτ(n)‖ = 0. (36)

Therefore, we haved(xτ(n), Sxτ(n)) ≤ ‖xτ(n) − vτ(n)‖ → 0 as n → ∞. (37)

Since {xτ(n)} is bounded, there exists a subsequence {uτ(n)} of {xτ(n)} that converges weakly to apoint x∗ ∈ H1. From ‖uτ(n) − xτ(n)‖ → 0, it follows that uτ(n) ⇀ x∗ ∈ H1.

Moreover, as in Case 1, we show that x∗ ∈ F(S) ∩Ω. Furthermore, since { fn(x)} is uniformlyconvergent on D ⊂ H1, we obtain that

lim supn→∞

〈 fτ(n)(p)− p, xτ(n)+1 − p〉 ≤ 0.

From (23), we get

‖xτ(n)+1 − p‖2 ≤ (1− 2ατ(n)(1− μ∗(1− ατ(n))) + α2

τ(n)(1 + μ∗2))‖xτ(n) − p‖2

− βτ(n)δτ(n)‖xτ(n) − zτ(n)‖2 + 2ατ(n)〈 fτ(n)(p)− p, xτ(n)+1 − p〉+ 2

(1− ατ(n)(1− μ∗(1− ξ))

)δτ(n)θτ(n)‖xτ(n) − xτ(n)−1‖‖xτ(n) − p‖

+ (1− ατ(n))δτ(n)θ2τ(n)‖xτ(n) − xτ(n)−1‖2

≤ (1− 2ατ(n)(1− μ∗)

)‖xτ(n) − p‖2 + α2τ(n)(1 + μ∗2)‖xτ(n) − p‖2

+ δτ(n)θn‖xτ(n) − xτ(n)−1‖(2(1− ατ(n)(1− μ∗))‖xτ(n) − p‖

+ (1− ατ(n))θτ(n)‖xτ(n) − xτ(n)−1‖)+ 2ατ(n)〈 fτ(n)(p)− p, xτ(n)+1 − p〉,

which implies that

2ατ(n)(1− μ∗)‖xτ(n) − p‖2 ≤ ‖xτ(n) − p‖2 − ‖xτ(n)+1 − p‖2 + α2τ(n)(1 + μ∗2)‖xτ(n) − p‖2

+ δτ(n)θn‖xτ(n) − xτ(n)−1‖(2(1− ατ(n)(1− μ∗))‖xτ(n) − p‖

+ (1− ατ(n))θτ(n)‖xτ(n) − xτ(n)−1‖)

+ 2ατ(n)〈 fτ(n)(p)− p, xτ(n)+1 − p〉,

or

2(1− μ∗)‖xτ(n) − p‖2 ≤ ατ(n)(1 + μ∗2)‖xτ(n) − p‖2 + 2〈 fτ(n)(p)− p, xτ(n)+1 − p〉

+ δτ(n)θτ(n)

ατ(n)‖xτ(n) − xτ(n)−1‖

(2(1− ατ(n)(1− μ∗))‖xτ(n) − p‖

+ (1− ατ(n))ατ(n)θτ(n)

ατ(n)‖xτ(n) − xτ(n)−1‖

).

276


Thus we havelim sup

n→∞‖xτ(n) − p‖ ≤ 0

and solim

n→∞‖xτ(n) − p‖ = 0. (38)

By (35) and (38), we get

‖xτ(n)+1 − p‖ ≤ ‖xτ(n)+1 − xτ(n)‖+ ‖xτ(n) − p‖ → 0, n → ∞.

Furthermore, for all n ≥ n0, it is easy to see that Γτ(n) ≤ Γτ(n)+1 if n �= τ(n) (that is, τ(n) < n)because of Γj ≥ Γj+1 for τ(n) + 1 ≤ j ≤ n. Consequently, it follows that, for all n ≥ n0,

0 ≤ Γn ≤ max{Γτ(n), Γτ(n)+1} = Γτ(n)+1.

Therefore, lim Γn = 0, that is, {xn} converges strongly to the point x∗. This completes the proof.

Remark 1. [22] The condition (C4) is easily implemented in numerical results because the value of ‖xn− xn−1‖is known before choosing θn. Indeed, we can choose the parameter θn such as

θn =

⎧⎨⎩min{

ω, ωn‖xn−xn−1‖

}, if ‖xn − xn−1‖ �= 0,

ω, otherwise,

where {ωn} is a positive sequence such that ωn = o(αn). Moreover, in the condition (C4), we can take

αn =1

n + 1, ω =

45

and

θn =

⎧⎨⎩min{

ω, α2n

‖xn−xn−1‖}

, if ‖xn − xn−1‖ �= 0,

ω, otherwise,

or

θn =

⎧⎨⎩min{

45 , 1

(n+1)2‖xn−xn−1‖}

, if ‖xn − xn−1‖ �= 0,45 , otherwise.

If the multi-valued quasi-nonexpansive mapping S in Theorem 1 is a single-valuedquasi-nonexpansive mapping, then we obtain the following:

Corollary 1. Let H1 and H2 be two real Hilbert spaces. Suppose that A : H1 → H2 is a bounded linear operatorwith adjoint operator A∗. Let { fn} be a sequence of μn-contractions fn : H1 → H1 with 0 < μ∗ ≤ μn ≤ μ∗ <1 and { fn(x)} be uniformly convergent for any x in a bounded subset D of H1. Suppose that S : H1 → H1 isa single-valued quasi-nonexpansive mapping, I − S is demiclosed at the origin and F(S) ∩Ω �= ∅. For anyx0, x1 ∈ H1, let the sequences {yn}, {un}, {zn} and {xn} be generated by⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩



λ − I)Ayn),

zn = ξSxn + (1− ξ)un,


(39)

277



λ −I)Ayn‖2


with 0 < τ∗ ≤ τn ≤ τ∗ < 1, {θn} ⊂ [0, ω) for


(C1) limn→∞ αn = 0;(C2) ∑∞



Then the sequence {xn} generated by (39) converges strongly to a point p ∈ F(S)∩Ω, where p = PF(S)∩Ω f (p).

Remark 2. If θn = 0, then the iterative scheme (14) in Theorem 1 reduces to the iterative (11).

4. Applications

In this section, we give some applications of the problem (SFPIP) in the variational inequalityproblem and game theory. First, we introduce variational inequality problem in [34] and game theory(see [35]).

4.1. The Variational Inequality Problem

Let C be a nonempty closed and convex subset of a real Hilbert space H1. Suppose that an operatorF : H1 → H1 is monotone.

Now, we consider the following variational inequality problem (VIP):

Find a point x∗ ∈ C such that 〈Fx∗, y− x∗〉 ≥ 0 for all y ∈ C. (40)

The solution set of the problem (VIP) is denoted by Γ.Moreover, it is well-known that x∗ is a solution of the problem (VIP) if and only if x∗ is a solution

of the problem (FPP) [34], that is, for any γ > 0,

x∗ = PC(x∗ − γFx∗).

The following lemma is extracted from [2,36]. This lemma is used for finding a solution of thesplit inclusion problem and the variational inequality problem:

Lemma 5. Let H1 be a real Hilbert space, F : H1 → H1 be a monotone and L-Lipschitz operator on a nonemptyclosed and convex subset C of H1. For any γ > 0, let T = PC(I − γF(PC(I − γF))). Then, for any y ∈ Γ andLγ < 1, we have

‖Tx− Ty‖ ≤ ‖x− y‖,

I − T is demiclosed at the origin and F(T) = Γ.

Now, we apply our Theorem 1, by combining with Lemma 5, to find a solution of the problem(VIP), that is, a point in the set Γ.

let B1 : H1 → 2H1 and B2 : H2 → 2H2 be maximal monotone mappings defined on H1 and H2,respectively, and A : H1 → H2 be a bounded linear operator with its adjoint A∗.

Now, we consider the split fixed point variational inclusion problem (SFPVIP) as follows:

Find a point x∗ ∈ H1 such that 0 ∈ B1(x∗), x∗ ∈ Γ (41)

andy∗ = Ax∗ ∈ H2 such that 0 ∈ B2(y∗). (42)

278


Theorem 2. Let H1 and H2 be two real Hilbert spaces, A : H1 → H2 be a bounded linear operator withits adjoint A∗. Let { fn} be a sequence of μn-contractions fn : H1 → H1 with 0 < μ∗ ≤ μn ≤ μ∗ < 1and { fn(x)} be uniformly convergent for any x in a bounded subset D of H1. For any λ > 0, let T =

PC(I − γF(PC(I − γF))) with Lγ < 1, where F : H1 → H1 is a L-Lipschitz and monotone operator onC ⊂ H1 and F(T) ∩ Ω �= ∅. For any x0, x1 ∈ H1, let the sequences {yn}, {un}, {zn} and {xn} begenerated by ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩



λ − I)Ayn),

zn = ξTxn + (1− ξ)un,


(43)


λ −I)Ayn‖2


with 0 < τ∗ ≤ τn ≤ τ∗ < 1, {θn} ⊂ [0, ω) for


(C1) limn→∞ αn = 0;(C2) ∑∞

n=1 αn = ∞;(C3) 0 < ε1 ≤ βn, 0 < ε2 ≤ δn;(C4) limn→∞


Then the sequence {xn} generated by (43) converges strongly to a point p ∈ F(T) ∩ Ω = Γ ∩ Ω, wherep = PΓ∩Ω f (p).

Proof. Since I − T is demiclosed at the origin and F(T) = Γ, by using Lemma (5) and Corollary (1),the sequence {xn} converges strongly to a point p ∈ F(T) ∩Ω, that is, the sequence {xn} convergesstrongly to a point p ∈ Γ.

4.2. Game Theory

Now, we consider a game of N players in strategic form

G = (pi, Si),

where i = 1, · · · , N, pi : S = S1 × S2 × · · · × SN → R is the pay-off function (continuous) of the ithplayer and Si ∈ RMi is the set of strategy of the ith player such that Mi = |Si|.

Let Si be nonempty compact and convex set, si ∈ Si be the strategy of the ith player ands = (s1, s2, · · · , sN) be the collective strategy of all players. For any s ∈ S and zi ∈ Si of the ith playerfor each i, the symbols S−i, s−i and (zi, s−i) are defined by

• S−i := (S1 × · · · × Si−1 × Si+1 × · · · × SN) is the set of strategies of the remaining players whensi was chosen by ith player,

• s−i := (s1, · · · , si−1, si+1, · · · , sN) is the strategies of the remaining players when ith player has siand

• (zi, s−i) := (s1, · · · , si−1, zi, si+1, · · · , sN) is the strategies of the situation that zi was chosen by ithplayer when the rest of the remaining players have chosen s−i.

Moreover, si is a special strategy of the ith player, supporting the player to maximize his pay-off,which equivalent to the following:

pi(si, s−i) = maxzi∈Si

pi(zi, s−i).

279


Definition 5. [37,38] Given a game of N players in strategic form, the collective strategies s∗ ∈ S is said to bea Nash equilibrium point if

pi(s∗) = maxzi∈Si

pi(zi, s∗i )

for all i = 1, · · · , N and s∗i ∈ S−i.

If no player can change his strategy to bring advantages, then the collective strategies s∗ = (s∗i , s∗−i)

is a Nash equilibrium point. Furthermore, a Nash equilibrium point is the collective strategies of allplayers, i.e., s∗i (for each i ≥ 1) is the best response of ith player. There is a multi-valued mappingTi : S−i → 2Si such that

Ti(s−i) = arg max pi(zi, s−i)

= {si ∈ Si : pi(si, s−i) = maxzi∈Si

pi(zi, s−i)}

for all s−i ∈ S−i. Therefore, we can define the mapping T : S → 2S by

T := T1 × T2 × · · · × TN

such that the Nash equilibrium point is the collective strategies s∗, where s∗ ∈ F(T). Note thats∗ ∈ F(T) is equivalent to s∗i ∈ T(s∗−i).

Let H1 and H2 be two real Hilbert spaces, B1 : H1 → 2H1 and B2 : H2 → 2H2 be multi-valuedmappings. Suppose S is nonempty compact and convex subset of H1 = RMN , H2 = R and the rest ofthe players have made their best responses s∗−i. For each s ∈ S, define a mapping A : S → H2 by

As = pi(s)− pi(zi, s∗−i),

where pi is linear, bounded and convex. Indeed, A is also linear, bounded and convex.

The Nash equilibrium problem (NEP) is the following:

Find a point s∗ ∈ S such that As∗ > 0, 0 ∈ H2. (44)

However, the solution to the problem (NEP) may not be single-valued. Then the problem (NEP)

reduces to finding the fixed point problem (FPP) of a multi-valued mapping, i.e.,

Find a point s∗ ∈ S such that s∗ ∈ Ts∗, (45)

where T is multi-valued pay-off function.Now, we apply our Theorem 1 to find a solution to the problem (FPP).

Let B1 : H1 → 2H1 and B2 : H2 → 2H2 be maximal monotone mappings defined on H1 and H2,respectively, and A : H1 → H2 be a bounded linear operator with its adjoint A∗.

Now, we consider the following problem:

Find a point s∗ ∈ H1 such that 0 ∈ B1(s∗), s∗ ∈ Ts∗ (46)

andy∗ = As∗ ∈ H2 such that 0 ∈ B2(y∗). (47)

Theorem 3. Assume that B1 and B2 are maximal monotone mappings defined on Hilbert spaces H1 and H2,respectively. Let T : S → CB(S) be a multi-valued quasi-nonexpansive mapping such that T is demiclosedat the origin. Let { fn} be a sequence of μn-contractions fn : H1 → H1 with 0 < μ∗ ≤ μn ≤ μ∗ < 1 and{ fn(x)} be uniformly convergent for any x in a bounded subset D of H1. Suppose that the problem (NEP) has

280


a nonempty solution and F(T) ∩Ω �= ∅. For arbitrarily chosen x0, x1 ∈ H1, let the sequences {yn}, {un},{zn} and {xn} be generated by ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩



λ − I)Ayn),

zn = ξvn + (1− ξ)un, vn ∈ Txn,


(48)


λ −I)Ayn‖2


with 0 < τ∗ ≤ τn ≤ τ∗ < 1, {θn} ⊂ [0, ω) for


(C1) limn→∞ αn = 0;(C2) ∑∞



Then the sequence {xn} generated by Equation (48) converges strongly to Nash equilibrium point.

Proof. By Theorem 1, the sequence {xn} converges strongly to a point p ∈ F(T)∩Ω, then the sequence{xn} converges strongly to a Nash equilibrium point.

Author Contributions: All five authors contributed equally to work. All authors read and approved the finalmanuscript. P.K. and K.S. conceived and designed the experiments. P.P., W.J. and Y.J.C. analyzed the data.P.P. and W.J. wrote the paper. Authorship must be limited to those who have contributed substantially to thework reported.

Funding: The Royal Golden Jubilee PhD Program (Grant No. PHD/0167/2560). The Petchra Pra Jom KlaoPh.D. Research Scholarship (Grant No. 10/2560). The King Mongkut’s University of Technology North Bangkok,Contract No. KMUTNB-KNOW-61-035.

Acknowledgments: The authors acknowledge the financial support provided by King Mongkut’s University ofTechnology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. Pawicha Phairatchatniyomwould like to thank the “Science Graduate Scholarship", Faculty of Science, King Mongkut’s University of TechnologyThonburi (KMUTT) (Grant No. 11/2560). Wachirapong Jirakitpuwapat would like to thank the Petchra PraJom Klao Ph.D. Research Scholarship and the King Mongkut’s University of Technology Thonburi (KMUTT)for financial support. Moreover, Kanokwan Sitthithakerngkiet was funded by King Mongkut’s University ofTechnology North Bangkok, Contract No. KMUTNB-KNOW-61-035.


References

1. Moudafi. Split monotone variational inclusions. J. Opt. Theory Appl. 2011, 150, 275–283. [CrossRef]2. Shehu, Y.; Agbebaku, D. On split inclusion problem and fixed point problem for multi-valued mappings.

Comput. Appl. Math. 2017, 37. [CrossRef]3. Shehu, Y.; Ogbuisi, F.U. An iterative method for solving split monotone variational inclusion and fixed point

problems. Rev. Real Acad. Cienc. Exact. Fís. Nat. Serie A. Mat. 2015, 110, 503–518. [CrossRef]4. Kazmi, K.; Rizvi, S. An iterative method for split variational inclusion problem and fixed point problem for

a nonexpansive mapping. Opt. Lett. 2013, 8. [CrossRef]5. Sitthithakerngkiet, K.; Deepho, J.; Kumam, P. A hybrid viscosity algorithm via modify the hybrid steepest

descent method for solving the split variational inclusion in image reconstruction and fixed point problems.Appl. Math. Comput. 2015, 250, 986–1001. [CrossRef]

6. Censor, Y.; Gibali, A.; Reich, S. Algorithms for the split variational inequality problem. Num. Algorithms2012, 59, 301–323. [CrossRef]

7. Byrne, C.; Censor, Y.; Gibali, A.; Reich, S. Weak and strong convergence of algorithms for the split commonnull point problem. Technical Report. arXiv 2011, arXiv:1108.5953.

281


8. Moudafi, A. The split common fixed-point problem for demicontractive mappings. Inverse Prob. 2010,26, 055007. [CrossRef]

9. Yao, Y.; Liou, Y.C.; Postolache, M. Self-adaptive algorithms for the split problem of the demicontractiveoperators. Optimization 2017, 67, 1309–1319. [CrossRef]

10. Dang, Y.; Gao, Y. The strong convergence of a KM-CQ-like algorithm for a split feasibility problem.Inverse Prob. 2011, 27, 015007. [CrossRef]

11. Sahu, D.R.; Pitea, A.; Verma, M. A new iteration technique for nonlinear operators as concerns convexprogramming and feasibility problems. Numer. Algorithms 2019. [CrossRef]

12. Censor, Y.; Elfving, T. A multiprojection algorithm using Bregman projections in a product space.Numer. Algorithms 1994, 8, 221–239. [CrossRef]

13. Combettes, P. The convex feasibility problem in image recovery. Adv. Imag. Electron. Phys. 1996, 95, 155–270.[CrossRef]

14. Kazmi, K.R.; Rizvi, S.H. Iterative approximation of a common solution of a split equilibrium problem, avariational inequality problem and a fixed point problem. J. Egypt. Math. Soc. 2013, 21, 44–51. [CrossRef]

15. Peng, J.W.; Wang, Y.; Shyu, D.S.; Yao, J.C. Common solutions of an iterative scheme for variational inclusions,equilibrium problems, and fixed point problems. J. Inequal. Appl. 2008, 15, 720371. [CrossRef]

16. Jung, J.S. Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces.Nonlinear Anal. Theory, Meth. Appl. 2007, 66, 2345–2354. [CrossRef]

17. Panyanak, B. Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces.Comput. Math. Appl. 2007, 54, 872–877. [CrossRef]

18. Shahzad, N.; Zegeye, H. On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces.Nonlinear Anal. 2009, 71, 838–844. [CrossRef]

19. Polyak, B. Some methods of speeding up the convergence of iteration methods. USSR Comput. Math.Math. Phys. 1964, 4, 1–17. [CrossRef]

20. Nesterov, Y. A method of solving a convex programming problem with convergence rate O(1/sqr(k)).Sov. Math. Dokl. 1983, 27, 372–376.

21. Dang, Y.; Sun, J.; Xu, H. Inertial accelerated algorithms for solving a split feasibility problem. J. Ind.Manag. Optim. 2017, 13, 1383–1394. [CrossRef]

22. Suantai, S.; Pholasa, N.; Cholamjiak, P. The modified inertial relaxed CQ algorithm for solving the splitfeasibility problems. J. Ind. Manag. Opt. 2017, 13, 1–21. [CrossRef]

23. Alvarez, F.; Attouch, H. An inertial proximal method for maximal monotone operators via discretization ofa nonlinear oscillator with damping. Wellposedness in optimization and related topics (Gargnano, 1999).Set-Valued Anal. 2001, 9, 3–11. [CrossRef]

24. Rockafellar, R.T. Monotone operators and the proximal point algorithm. SIAM J. Control Opt. 1976,14, 877–898. [CrossRef]

25. Attouch, H.; Peypouquet, J.; Redont, P. A dynamical approach to an inertial forward-backward algorithmfor convex minimization. SIAM J. Opt. 2014, 24, 232–256. [CrossRef]

26. Bot, R.I.; Csetnek, E.R. An inertial alternating direction method of multipliers. Min. Theory Appl. 2016,1, 29–49.

27. Maingé, P.E. Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 2008,219, 223–236. [CrossRef]

28. Bot, R.I., Csetnek, E.R., Hendrich, C. Inertial Douglas-Rachford splitting for monotone inclusion problems.Appl. Math. Comput. 2015, 256, 472-–487.

29. Chuang, C.S. Strong convergence theorems for the split variational inclusion problem in Hilbert spaces.Fix. Point Theory Appl. 2013, 2013. [CrossRef]

30. Che, H.; Li, M. Solving split variational inclusion problem and fixed point problem for nonexpansivesemigroup without prior knowledge of operator norms. Math. Prob. Eng. 2015, 2015, 1–9. [CrossRef]

31. Ansari, Q.H.; Rehan, A.; Wen, C.F. Split hierarchical variational inequality problems and fixed point problemsfor nonexpansive mappings. J. Inequal. Appl. 2015, 16, 274. [CrossRef]

32. Xu, H.K. Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66, 240–256. [CrossRef]33. Maingé, P.E. Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces.

J. Math. Anal. Appl. 2007, 325, 469–479. [CrossRef]

282


34. Glowinski, R.; Tallec, P. Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics;SIAM Studies in Applied Mathematics; Society for Industrial and Applied Mathematics: Philadelphia,PA, USA, 1989.

35. Von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press:Princeton, NJ, USA, 1947.

36. Kraikaew, R.; Saejung, S. Strong convergence of the Halpern subgradient extragradient method for solvingvariational inequalities in Hilbert spaces. J. Opt. Theory Appl. 2014, 163, 399–412. [CrossRef]

37. Nash, J.F., Jr. Equilibrium points in n-person games. Proc. Nat. Acad. Sci. USA 1950, 36, 48–49. [CrossRef][PubMed]

38. Nash, J. Non-cooperative games. Ann. Math. 1951, 54, 286–295. [CrossRef]


283

mathematics

Article

A Fast Derivative-Free Iteration Scheme for NonlinearSystems and Integral Equations

Mozafar Rostami, Taher Lotfi * and Ali Brahmand

Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan 15743-65181, Iran* Correspondence: [email protected]

Received: 23 June 2019; Accepted: 11 July 2019; Published: 18 July 2019

Abstract: Derivative-free schemes are a class of competitive methods since they are one remedy in casesat which the computation of the Jacobian or higher order derivatives of multi-dimensional functionsis difficult. This article studies a variant of Steffensen’s method with memory for tackling a nonlinearsystem of equations, to not only be independent of the Jacobian calculation but also to improve thecomputational efficiency. The analytical parts of the work are supported by several tests, including anapplication in mixed integral equations.

Keywords: integral equation; efficiency index; nonlinear models; iterative methods; higher order

MSC: 65H10; 45D05

1. Introductory Notes

1.1. Background

There exist many works handling the approximate solution of linear and nonlinear integral equations.However, tackling nonlinear integral equations would be more challenging due to the presence ofnonlinearity which might be expensive for different solvers [1,2].

Some authors discussed the asymptotic error expansion of collocation-type and Nystrom-typemethods for Volterra–Fredholm integral equations with nonlinearity, see [3] for a complete discussion onthis issue. One class of nonlinear internal equations is the mixed Hammerstein integral equations withseveral application in engineering problems [2].

Since in the process of finding the solution of such integral equations, most of the time a systemof algebraic equation would occur that must be solved quickly and accurately, thus we here bring theattention to develop and study a useful numerical solution scheme for solving nonlinear systems withapplication in tackling nonlinear integral equations.

Clearly, there are some other nonlinear problems in literature which could yield in tackling nonlinearsystem of equations, see e.g., [4,5].

Mathematics 2019, 7, 637; doi:10.3390/math7070637 www.mdpi.com/journal/mathematics

284


1.2. Definition

Consider a nonlinear system of equations of algebraic type as follows [6]:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩a1(x1, x2, . . . , xm) = 0,a2(x1, x2, . . . , xm) = 0,...

am(x1, x2, . . . , xm) = 0,

(1)

which contains m equations with m unknowns and A(x) = (a1(x), a2(x), . . . , am(x))T while a1(x), a2(x),. . ., am(x) are the functions of coordinate. We can also write (1) using x = (x1, x2, . . . , xm) in a morecompact form as

A(x) = 0. (2)

The purpose of this work is to study finding the solution of system (1) via iteration process anddiscuss its application in solving nonlinear integral equations. As such, now let us briefly review some ofthe existing methods for finding its simple roots in the next subsection.

1.3. Existing Solvers

The Steffensen’s scheme for solving nonlinear systems is written as follows [7]:{w(n) = x(n) + A(x(n)), x(0) ∈ Rm,x(n+1) = x(n) − [x(n), w(n); A]−1 A(x(n)), n = 0, 1, 2, · · · ,

(3)

which is based upon the divided difference operator (DDO). The 1st order DDO of A for themultidimensional nodes x and y is expressed by a component-to-component procedure as follows [8]:

[x, y; A]i,j =Ai(x1, . . . , xj, yj+1, . . . , ym)− Ai(x1, . . . , xj−1, yj, . . . , ym)

xj − yj, 1 ≤ i, j ≤ m. (4)

Recall that first-order divided difference of A on Rm is a mapping as follows:

[·, ·; A] : D ⊂ Rm ×R

m → L(Rm), (5)

that reads[y, x; A](y− x) = A(y)− A(x), ∀x, y ∈ D. (6)

Here L(·) shows the set of bounded linear functions. By considering h = y− x, one we can alsoexpress the first-order DDO as follows [8]:

[x + h, x; A] =∫ 1

0A′(x + th)dt, ∀(x, h) ∈ R

m ×Rm. (7)

Traub in [9] investigated another way based on the function J(x, H) for approximating the Jacobianmatrix of the Newton’s method and to obtain the Steffensen’s scheme based on a point-wise definition.

An improvement of (3) was given in [10,11] as follows:{z(n) = x(n) − [x(n), w(n); A]−1 A(x(n)),x(n+1) = z(n) − [x(n), w(n); A]−1 A(z(n)),

(8)

285


whereinw(n) = x(n) + βA(x(n)), β ∈ R. (9)

The point in (8) in contrast to (3) is that it applies two steps and of course two m-D functionalevaluations to reach a higher rate than quadratic. Here, the idea is to freeze the DDO per cycle andthen increase the sub steps so as to gain as much as possible of order improvement, as well as someimprovements in the computational efficiency index of the scheme.

Let us also recall some of the iteration schemes having the requirement of Jacobian computation now.The Jarratt’s iteration having fourth rate of convergence for solving (1) is given by [12]:⎧⎪⎨⎪⎩

z(n) = x(n) − 23 A′(x(n))−1 A(x(n)),

x(n+1) = x(n) − 12 (3A′(z(n))− A′(x(n)))−1

×(3A′(z(n)) + A′(x(n)))A′(x(n))−1 A(x(n)).(10)

This fourth-order iteration expression requires the computation of two matrix inverses (based on theresolution of linear systems) to achieve its rate, which manifest that getting higher rate of convergence inthe form of a two-step method is costly.

1.4. Motivation

All methods discussed until now are without memory; some improvements over such schemes canbe done by considering additional memory terms.

Our motivation of pursuing this aim is not only limited at tackling nonlinear systems, but a motivationis to apply such schemes for practical engineering problems such as the nonlinear mixed integral equations,see e.g., [13–16].

The goal in our development is to reach a higher computational efficiency using as low as possiblenumber of linear systems of equations and the functional evaluations. This is directly interlinked with theconcept of scientific computing and numerical analysis which gives a meaning to the investigation andproposing novel numerical procedures.

1.5. Achievement and Contribution

The objective of this work is to present a two-step higher order scheme to solve system of nonlinearequations. As such, we present an iteration method with memory for finding both real and complex zeros.Our scheme does not require computing the Fréchet derivatives of the function.

1.6. Organization

We unfold this article as follows. In Section 2, the derivation and contribution of an iteration expressionis furnished. Section 3 provides an error analysis for its convergence rate. The computational efficiency ofdifferent solvers by including not only the number of functional evaluations, but also the number of systemof involved linear equations, the number of LU (Lower-Upper) factorizations as well as the other similaroperations will be discussed in Section 4 in detail. Section 5 discusses the application of the proposedscheme. Concluding remarks are given in Section 6.

2. A Derivative-Free Scheme

Here our attempt is to increase the computational efficiency index of (8) without imposing severalmore steps or further DDDs in each cycle. To complete such a task, we rely on the concept of methods withmemory which state that the convergence speed and efficiency of iterative methods could be improved bysaving and using the already computed values of the functions and nodes.

286


In fact, the error equation of the uni-parametric family of methods (8) includes a term of theform below:

I + βA′(α) = 0. (11)

The free nonzero parameter β in (11) can clearly affect not only on the domain of convergence(attraction basins of the iterative method) but also to the improvement of the convergence order.When tackling a nonlinear system of equations, and since α is not known, we can use an approximationfor A′(α) to make the whole relation (11) approximately zero. Therefore, we may write

β � −A′(α)−1, (12)

wherein α is an approximation of the solution (per cycle).It is important to discuss how we approximate the matrix β := B(n)(n ≥ 1) by employing some

estimates to −A′(α) computed via the existing data.To improve the performance of (8) using the notion of methods with memory, we consider the

following iteration expression:⎧⎪⎨⎪⎩w(n) = x(n) + βA(x(n))z(n) = x(n) − [x(n), w(n); A]−1 A(x(n)),

x(n+1) = z(n) − [x(n), w(n); A]−1 A(z(n)).(13)

To ease up the implementation of the scheme with memory, let us first consider

β := B(n) = −[w(n−1), x(n−1); A]−1 = −M(n−1)−1 ≈ −A′(α)−1. (14)

and {M(n−1)δ(n) = A(x(n)),M(n−1)γ(n) = A(y(n)).

(15)

Thus, now we contribute the following scheme:⎧⎪⎪⎪⎨⎪⎪⎪⎩B(n) = −[w(n−1), x(n−1); A]−1, n ≥ 1,w(n) = x(n) + B(n)A(x(n)), n ≥ 1,y(n) = x(n) + δ(n), n ≥ 0,x(n+1) = y(n) + γ(n).

(16)

Lemma 1. Let D ⊂ Rm be a nonempty convex domain. Suppose that A is thrice Fréchet differentiable on D,and that [u, v; A] ∈ L(D, D), for any u, v ∈ D(u �= v) and the initial value x(0) and the solution α are closeto each other. By considering B(n) = −[w(n−1), x(n−1); A]−1 and d(n) := I + B(n) A′(α), one obtains the errorrelation below

d(n) ∼ e(n−1). (17)

Proof. See [17] for more details.

To implement (16), one needs to solve some linear systems of algebraic equations. This means that ateach new step, a new LU factorization is needed, and no information can be exploited from the previoussteps. However, there exists a body of literature about recycling this kind of information to obtain updatedpreconditioner for iterative solvers, [18–20]. We leave future discussion about constructing and imposingsuch a preconditioner for future works in this field.

287


As long as the coefficient matrices are sparse or large and sparse, a Krylov subspace method can beemployed to speed up the process. However, the merit in (16) is that the two linear systems have one samecoefficient matrix. Hence, only one LU factorization would be enough and by saving the decomposition,one can act it to two different right-hand-side vectors to get the solution vectors per sub cycles of (16).

A challenging part of the implementation using (16) is the incorporation of B(n). This is a not anymorea constant and it should be defined as a matrix. In this paper, whenever required the initial matrix B(0) isspecified by:

B(0)−1= diag

(− 1

1000

). (18)

The choice of the initial matrix B(0) affects directly on the whole process in order to arrive in theconvergence phase as quickly as possible. Here, (18) is in agreement with the dynamical studies ofSteffensen-type methods with memory at which the basins of attractions are larger as long as the freeparameter is close to zero.

Noting also that updating B(n) per cycle is again based on the already computed LU factorizationwhile it should only act on the identity matrix to proceed.

3. Rate of Convergence

It is known that via the Taylor expansion of A′(x + th) in the node x and integrating, one obtains:

∫ 1

0A′(x + th)dt = A′(x) +

12

A′′(x)h +16

A′′′(x)h2 +1

24A(iv)(x)h3 +O(h4). (19)

It is here assumed that A′(α) is not singular and e(n) = x(n) − α is called the error at the n-th iterateand [6,21]:

e(n+1) = He(n)p+O(e(n)

p+1), (20)

(20) is the equation of error, whereas H is a p-linear function. This means that H ∈ L(Rm,Rm, . . . ,Rm).Moreover, we consider:

e(n)p= (e(n), e(n), . . . , e(n)︸︷︷︸

p times

), (21)

which would be a matrix.Before stating the main theorem, it is pointed out that if A be differentiable in terms of Fréchet concept

in D sufficiently. As in [22], the l-th differentiation of A at u ∈ Rm, l ≥ 1, is the following l-linear function

A(l)(u) : Rm × · · · ×Rm −→ R

m, (22)

so that A(l)(u)(v1, v2, . . . , vl) ∈ Rm. It is also famous that, for α + h ∈ Rm locating in a neighborhood of aroot α of (1), the Taylor expansion could be written and we have [22]:

A(α + h) = A′(α)[

h +p−1

∑l=2

Clhl

]+O(hp), (23)

whereinCl = (1/l!)[A′(α)]−1 A(l)(α), l ≥ 2. (24)

288


One finds Clhl ∈ Rm, because A(l)(α) ∈ L(Rm × · · · ×Rm,Rm) and [A′(α)]−1 ∈ L(Rm). Moreover,for A′ we have:

A′(α + h) = A′(α)[

I +p−1

∑l=2

lClhl−1

]+O(hp), (25)

where I is the unit matrix of appropriate size. Here, lClhl−1 ∈ L(Rm).

Theorem 1. Assume that in (1), A : D ⊆ Rm −→ Rm is Fréchet differentiable sufficiently at any points of D atα ∈ Rm. Here we assume A(α) = 0 and det(A′(x)) �= 0. Then, (16) with a choice of suitable initial vector has 3.30R-order of convergence.

Proof. For the iteration scheme (16) in the case without memory and using (23)–(25), we can obtain:

e(n+1) = (βA′(α) + I)(βA′(α) + 2I)C22e(n)

3+O(e(n)

4). (26)

Let us now re-write (26) in the asymptotical form as comes next:

e(n+1) ∼ d(n)1 e(n)3. (27)

Several symbolic calculations by taking into account that the coefficient of the error terms in our m-Dcase are all matrices, Lemma 1, and their multiplications does not admit commutativity, one obtains that:

d(n)1 ∼ e(n−1), ∀n ≥ 1. (28)

Therefore, one attains:

d(n)1

2 ∼ e(n−1)2, ∀n ≥ 0. (29)

Combining (28) and (29) into (27), we attain:

e(n+1) ∼ e(n−1)1e(n)

3. (30)

It shows that1p+ 3 = p, (31)

wherein its convergence r-order is given by:

p =12

(√13 + 3

)� 3.30278. (32)

The proof is ended.

4. Efficiency

Here we only need to compute one LU factorization per cycle and act it two times for different linearsystems with two different right hand sides and one time on an identity matrix for the acceleration matrixB(n) to achieve a higher speed rate.

It is recalled that the classical index of efficiency is defined by [8]:

E = p1C , (33)

289


wherein p is the convergence rate and C is the whole burden per cycle considering the number offunctional evaluations.

When dealing with nonlinear system of equations, the cost of functional evaluations per cycle can beexpressed as follows:

• To evaluate A, m evaluations of functions are required.• To evaluate the associated Jacobian matrix A′ needs m2 evaluations of functions.• To evaluate the first-order DDO, we need m2 −m evaluations of functions.• In addition, the LU factorization cost is θ(2 m3

3 ) plus θ(2m2) in tackling the two involvedtriangular systems.

wherein θ is a weight that connects the cost of 1 evaluation of function with one flops. Here it is assumedthat θ = 1. No preconditioning is imposed in each cycle of these methods for solving the linear systems.This is done for all the compared methods.

To be more precise, we consider that the cost for computing each of the scalar functions is unit.The cost for computing other involved calculations are all also a factor of this unity cost. This is the way togive a flops-like efficiency index [23].

Considering only the consumed functional evaluations per cycle might not be a key element forreporting the indices of efficiency when solving nonlinear systems of equations. The number of matrixproducts, scalar products, decomposition of LU and the solution of the triangular systems of algebraiclinear equations are significant in estimating the real cost and superiority of a scheme in comparison to theexisting solvers in literature [23].

Hence, the results can be summarized as follows for large m:

21

2m33 +4m2

< 21

2m33 +3m2+m < 3

12m3

3 +6m2+m < 3.301

2m33 +6m2+m . (34)

In our comparisons, we applied the Newton’s quadratically convergent iteration expression (NM)and also Steffensen’s method (SM), the third-order expression of Amat et al. (8) denoted by AM, and thepresented approach (16) showed by PM, for tackling our nonlinear systems of algebraic equations. This isalso plotted in Figure 1 showing the competitiveness of the scheme with memory (16).

Figure 1. The comparison of flops-like efficiency indices for various schemes with and without memory bychanging m.

5. Computational Tests

The aim of this section is to reveal the application of our proposed nonlinear solver for some practicalproblems. The software Mathematica 11.0, [24,25] was used for doing all calculations regarding thecompared methods. We avoided computing any matrix inverse and the linear systems were solved

290


applying the command LinearSolver[]. For the implementation of such schemes a possible stoppingcriterion can be defined based on the residual norm and imposed as follows:

||A(x(k))||2 ≤ ε, (35)

wherein ε is the required accuracy. || · ||2 is the l2 norm.To confirm the theoretical convergence speed in our numerical tests, we obtain the the numerical rate

of convergence by employing the following definition:

ρ ≈ ln(||A(x(k+1))||2/||A(x(k))||2)ln(||A(x(k))||2/||A(x(k−1))||2) . (36)

5.1. An Academical Test

Example 1. Here a nonlinear system of equations A(x) = 0, having complex root is examined as comes next:

A(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

5 exp (x1 − 2)x2 + 2x7x10 + 8x3

x4 − 5x63 − x9,

5 tan (x1 + 2) + cos (x9x10) + x2

3 + 7x34 − 2 sin3 (x6),

x12 − x10x5x6x7x8x9 + tan (x2) + 2x3

x4 − 5x63,

2 tan (x12) + 2x2 + x3

2 − 5x53 − x6 + x8

cos (x9),10x1

2 − x10 + cos (x2) + x32 − 5x6

3 − 2x8 − 4x9 ,cos−1(x1

2) sin (x2)− 2x10x54x6x9 + x3

2,x1x2

x7 − x8x10 + x3

5 − 5x53 + x7,

cos−1 (−10x10 + x8 + x9) + x4 sin (x2) + x3 − 15x52 + x7,

10x1 + x32 − 5x5

2 + 10x6x8 − sin (x7) + 2x9,

x1 sin (x2)− 2x10x8 + x10 − 5x6 − 10x9,

(37)

wherein the exact solution just shown up to 10 decimal places as follows:

α � (1.3273490437 + 0.3502924960i, 1.058599346− 1.748724664i,1.0276186794− 0.0141308051i, 3.273950008 + 0.127828308i,0.8318243937 + 0.0017551949i,−0.4853245912 + 0.6848776400i,0.1693667630 + 0.1840917580i, 1.534419958− 0.321214766i,2.086379651 + 0.426342755i,−1.989592331 + 1.478395393i)∗.

(38)

The numerical evidences and the computational order of convergence ρ for this experiment arereported forward in Table 1 using 1000 fixed floating point arithmetic and the starting value x(0) = (1.2 +0.3I, 1.1− 1.9I, 1.0− 0.1I, 2.5 + 0.5I, 0.8− 0.1I,−0.4 + 1.I, 0.1 + 0.1I, 1.3− 0.7I, 2.0 + 0.5I,−1.9 + 1.4I)∗.Here, the residual norm ‖ · ‖2 is reported.

Table 1. Comparison evidences for Example 1.

Met. ‖A(x(3))‖ ‖A(x(4))‖ ‖A(x(5))‖ ‖A(x(6))‖ ‖A(x(7))‖ ‖A(x(8))‖ ‖A(x(9))‖ ρ

NM 8.19E− 1 2.73E− 2 1.79E− 5 1.28E− 11 2.52E− 23 8.28E− 47 2.50E− 94 2.02SM 7.68E− 1 1.83E− 2 7.33E− 6 5.17E− 12 1.51E− 24 2.03E− 49 3.91E− 99 1.99AM 8.50E− 2 6.50E− 6 4.98E− 17 3.68E− 50 1.26E− 149 4.83E− 448 3.00PM 6.80E− 2 1.12E− 7 5.15E− 26 1.76E− 86 6.47E− 287 3.31

291


5.2. An Integral Equation Using a Collocation Approach

Example 2. The purpose of this test was to examine the performance of the new derivative-free scheme with memoryfor the following mixed Hammerstein integral equation [6]:

y(s) = 1 +15

∫ 1

0G(s, t)y(t)3dt, (39)

wherein y ∈ C[0, 1], s, t ∈ [0, 1] and the kernel G is defined as follows:

G(s, t) =

{(1− s)t, t ≤ s,

s(1− t), t > s.(40)

By employing the well-resulted Gauss-Legendre quadrature formula in discretization of integralequations given in the following form, we will be able to tackle (39):

∫ 1

0y(t)dt ≈

χ

∑j=1

wjy(tj), (41)

where the abscissas tj and the weights wj were determined via the formula of Gauss–Legendre quadrature.The lower limit of integration in standard Gauss–Legendre quadrature formula is −1. In order to

approximate the integral (41) over [0, 1], we should map the roots of Legendre polynomials tj on thissegment and scale the weights wj.

Showing the estimation y(ti) via xi (i = 1, 2, · · · , χ), one would be able to transfigure the process ofsolving nonlinear mixed integral equations into a set of nonlinear algebraic equations as comes next:

5xi − 5−χ

∑j=1

cijx3j = 0, i = 1, 2, · · · , χ, (42)

where

cij =

{wjtj(1− ti), i f j ≤ i,wjti(1− tj), i f i < j.

(43)

For this example, we employed 200 digits floating point in the computations with the stop terminationas the following residual norm (35) with ε = 10−100. The initial vector was selected as x(0) = (3, 3, . . . , 3)∗,while the results are shown in Figure 2 using χ = 40 as a list log plot of the function values by performingthe cycle. It reveals a stable and fast performance of the new scheme with memory in solving integralequations. Recalling that the Figure 2 can be interpreted only as an error of numerical solution of thesystem (42) but not the error of solution of the source integral Equation (39).

Figure 2. Error history for solving the integral equation in Example 2 using PM (Performed by Mathematica).

292


6. Summary

For derivative-involved iteration schemes in solving nonlinear systems, we use the m×m Jacobianmatrix, i.e., F′(x), with entries F′(x)jk = ∂xk fj(x). Higher order schemes, such as Chebyshev methods,need higher multi dimensional derivatives which make them less practical. To be more precise, the firstFréchet derivative is a matrix with m2 elements, while the 2nd order Fréchet differentiation has m3 entries(ignoring the symmetric feature).

In this work, we have developed and introduced a variant of Steffensen’s method with memory fortackling nonlinear problems. The scheme consists of two steps and requires the the computation of onlyone LU factorization which makes its computational efficiency index higher than some of the existingsolvers in the literature.

The application of the iteration scheme for nonlinear integral equations via the collocation approachwas discussed and its application for other types of nonlinear discretized set of equations obtained frompractical problems such as the ones in [26,27] can be investigated similarly.

Author Contributions: All authors contributed equally in preparing and writing this work.

Funding: This work was supported by Hamedan Branch of Islamic Azad university.

Acknowledgments: We are grateful to three anonymous referees for several comments which improved the readabilityof this work.


References

1. Qasim, S.; Ali, Z.; Ahmad, F.; Serra-Capizzano, S.; Ullah, M.Z.; Mahmood, A. Solving systems of nonlinearequations when the nonlinearity is expensive. Comput. Math. Appl. 2016, 71, 1464–1478. [CrossRef]

2. Wazwaz, A.-M. Linear and Nonlinear Integral Equations; Higher Education Press: Beijing, China;Springer: Berlin/Heidelberg, Germany, 2011.

3. Mashayekhi, S.; Razzaghi, M.; Tripak, O. Solution of the nonlinear mixed Volterra-Fredholm integral equationsby hybrid of block-pulse functions and Bernoulli polynomials. Sci. World J. 2014, 2014. [CrossRef] [PubMed]

4. Alzahrani, E.O.; Al-Aidarous, E.S.; Younas, A.M.M.; Ahmad, F.; Ahmad, S.; Ahmad, S. A higher order frozenJacobian iterative method for solving Hamilton-Jacobi equations. J. Nonlinear Sci. Appl. 2016, 9, 6210–6227.[CrossRef]

5. Soleymani, F. Pricing multi–asset option problems: A Chebyshev pseudo–spectral method. BIT Numer. Math.2019, 59, 243–270. [CrossRef]

6. Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press:New York, NY, USA, 1970.

7. Noda, T. The Steffensen iteration method for systems of nonlinear equations. Proc. Jpn. Acad. 1987, 63, 186–189.[CrossRef]

8. Grau-Sánchez, M.; Grau, À.; Noguera, M. On the computational efficiency index and some iterative methods forsolving systems of nonlinear equations. J. Comput. Appl. Math. 2011, 236, 1259–1266. [CrossRef]

9. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice Hall: New York, NY, USA, 1964.10. Amat, S.; Busquier, S. Convergence and numerical analysis of a family of two-step Steffensen’s methods.

Comput. Math. Appl. 2005, 49, 13–22. [CrossRef]11. Soleymani, F.; Sharifi, M.; Shateyi, S.; Haghani, F.K. A class of Steffensen-type iterative methods for nonlinear

systems. J. Appl. Math. 2014, 2014. [CrossRef]12. Babajee, D.K.R.; Dauhoo, M.Z.; Darvishi, M.T.; Barati, A. A note on the local convergence of iterative methods

based on Adomian decomposition method and 3-node quadrature rule. Appl. Math. Comput. 2008, 200, 452–458.[CrossRef]

293


13. Alaidarous, E.S.; Ullah, M.Z.; Ahmad, F.; Al-Fhaid, A.S. An efficient higher-order quasilinearization method forsolving nonlinear BVPs. J. Appl. Math. 2013, 2013. [CrossRef]

14. Hanaç, E. The phase plane analysis of nonlinear equation. J. Math. Anal. 2018, 9, 89–97.15. Hasan, P.M.A.; Sulaiman, N.A. Numerical treatment of mixed Volterra-Fredholm integral equations using

trigonometric functions and Laguerre polynomials. ZANCO J. Pure Appl. Sci. 2018, 30, 97–106.16. Qasim, U.; Ali, Z.; Ahmad, F.; Serra-Capizzano, S.; Ullah, M.Z.; Asma, M. Constructing frozen Jacobian iterative

methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopymethod. Algorithms 2016, 9, 18. [CrossRef]

17. Ahmad, F.; Soleymani, F.; Khaksar Haghani, F.; Serra-Capizzano, S. Higher order derivative-free iterativemethods with and without memory for systems of nonlinear equations. Appl. Math. Comput. 2017, 314, 199–211.[CrossRef]

18. Bellavia, S.; Bertaccini, D.; Morini, B. Nonsymmetric preconditioner updates in Newton-Krylov methods fornonlinear systems. SIAM J. Sci. Comput. 2011, 33, 2595–2619. [CrossRef]

19. Bellavia, S.; Morini, B.; Porcelli, M. New updates of incomplete LU factorizations and applications to largenonlinear systems. Optim. Methods Softw. 2014, 29, 321–340. [CrossRef]

20. Bertaccini, D.; Durastante, F. Interpolating preconditioners for the solution of sequence of linear systems.Comput. Math. Appl. 2016, 72, 1118–1130. [CrossRef]

21. Sharma, J.R.; Kumar, D.; Argyros, I.K.; Magreñán, Á.A. On a bi-parametric family of fourth order compositeNewton-Jarratt methods for nonlinear systems. Mathematics 2019, 7, 492. [CrossRef]

22. Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. A modified Newton-Jarratt’s composition.Numer. Algorithms 2010, 55, 87–99. [CrossRef]

23. Montazeri, H.; Soleymani, F.; Shateyi, S.; Motsa, S.S. On a new method for computing the numerical solution ofsystems of nonlinear equations. J. Appl. Math. 2012, 2012, 1–15. [CrossRef]

24. Sánchez León, J.G. Mathematica Beyond Mathematics: The Wolfram Language in the Real World; Taylor & FrancisGroup: Boca Raton, FL, USA, 2017.

25. Wagon, S. Mathematica in Action, 3rd ed.; Springer: Berlin, Germany, 2010.26. Soheili, A.R.; Soleymani, F. Iterative methods for nonlinear systems associated with finite difference approach in

stochastic differential equations. Numer. Algorithms 2016, 71, 89–102. [CrossRef]27. Soleymani, F.; Barfeie, M. Pricing options under stochastic volatility jump model: A stable adaptive scheme.

Appl. Numer. Math. 2019, 145, 69–89. [CrossRef]

c© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution (CCBY) license (http://creativecommons.org/licenses/by/4.0/).

294

mathematics

Article

A Unified Convergence Analysis for Some Two-PointType Methods for Nonsmooth Operators

Sergio Amat 1,*,†, Ioannis Argyros 2,†, Sonia Busquier 1,†, Miguel Ángel Hernández-Verón 3,†

and María Jesús Rubio 3,†

1 Departamentode Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena,11003 Cádiz, Spain

2 Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA3 Departamento de Matemáticas y Computación, Universidad de La Rioja, Calle Madre de Dios, 53,

26006 Logrono, Spain* Correspondence: [email protected]; Tel.: +34-968-325-651† These authors contributed equally to this work.

Received: 18 June 2019; Accepted: 31 July 2019; Published: 3 August 2019

Abstract: The aim of this paper is the approximation of nonlinear equations using iterative methods.We present a unified convergence analysis for some two-point type methods. This way we comparespecializations of our method using not necessarily the same convergence criteria. We consider bothsemilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and inthe second on the solution. The results can be applied for smooth and nonsmooth operators.

Keywords: iterative methods; nonlinear equations; Newton-type methods; smooth andnonsmooth operators

1. Introduction

One of the most important techniques in order to approximate nonlinear equations are iterativemethods [1–6]. In this paper, we present a unified approach for two-point Newton-type methods forsmooth and nonsmooth operators [7–10]. We will consider two types of convergence. The semilocalconvergence is where the hypotheses are imposed on the initial guess; and local convergence is wherethe hypotheses are imposed on the solution. Our family includes a great variety of methods. We areinterested also in the application of these methods in practice (nonlinear systems, boundary problemsand image processing).

For a greater generality, in this study, let X and Y be two Banach spaces and D a nonempty,open, and convex set; let F1 : D ⊂ X → Y and F2 : D ⊂ X → Y be continuous operators. Moreover,we assume that the operator F1 has a continuous Fréchet derivative and F2 is a continuous operatorwhose differentiability is not assumed. We consider the equation

F(x) = F1(x) + F2(x) = 0. (1)

To solve this equation, we use the two-point Newton-type methods defined by

xk+1 = xk − L−1k−1,k(F1(xk) + F2(xk)) (2)

for each k = 0, 1, 2, ..., where x−1, x0 ∈ D are the initial points, L(., .) : D× D → L(X, Y) and L(X, Y)is the space of bounded linear operators from X into Y. We have denoted by Lk−1,k = L(xk−1, xk).

If F2(x) �= 0, we have that the operator F is not Fréchet differentiable. In general, to approximatea solution of (1) in this situation, derivative-free iterative methods are used [11–14]. To obtain this



type of iterative processes, it is common to approximate derivatives by difference divided. Rememberthat, given an operator H : D ⊂ X → Y, we call [x, y; H] ∈ L(X, Y) a first order divided differencesoperator for H on the points x and y (x �= y) in D if

[x, y; H](x− y) = H(x)− H(y). (3)

So, to solve (1) with iterative methods given from (2), we can consider at least two different procedures.Firstly, we have the Zincenko method [15], given by the following algorithm:⎧⎨⎩ Given x−1, x0 ∈ D,

xk+1 = xk −[F′1(xk)

]−1F(xk), n ≥ 0,(4)

where we directly eliminate the nondifferentiable part of F, i.e., F2. So, in this case, Lk−1,k = F′1(xk)

in (2). Secondly, we can consider an approximation of F′ by divided differences, the secant-typemethods [16,17]: ⎧⎪⎪⎨⎪⎪⎩

Given x−1, x0 ∈ D,

yk = λxk + (1− λ)xk−1, λ ∈ [0, 1),

xk+1 = xk − [yk, xk; F]−1F(xk), n ≥ 0,

(5)

where the secant method, for λ = 0, is obtained. So, in this case, Lk−1,k = [yk, xk; F] in (2). But, if weconsider a better approximation of the derivative of F, an approximation of second order, we have theKurchatov method [18]:⎧⎨⎩ Given x−1, x0 ∈ D,

xk+1 = xk − [xk−1, 2xk − xk−1; F]−1F(xn), n ≥ 0,(6)

in this case, Lk−1,k = [xk−1, 2xk − xk−1; F] in (2).By using this procedure of decomposition for operator F, we see that we can also consider the

application of iterative methods that use derivatives when F is nondifferentiable. So, if we considerdecomposition of F given in (1), we can use the Newton-secant-type algorithm:⎧⎪⎪⎪⎨⎪⎪⎪⎩

Given x−1, x0 ∈ D,

yk = λxk + (1− λ)xk−1, λ ∈ [0, 1),

xk+1 = xk −(

F′1(xk) + [yk, xk; F])−1F(xk), n ≥ 0,

(7)

where Lk−1,k = F′1(xk) + [yk, xk; F] in (2). The other possibility, from the decomposition method, is toconsider the Newton–Kurchatov [19] algorithm:⎧⎨⎩ Given x−1, x0 ∈ D,

xk+1 = xk −(

F′1(xk) + [xk−1, 2xk − xk−1; F])−1F(xk), n ≥ 0,

(8)

where Lk−1,k = F′1(xk) + [xk−1, 2xk − xk−1; F] in (2). Another possibility is to consider Steffensen-typemethods, that is, the methods associated to divided differences like [xk, xk + F(xk); F].

As we can see, there are a lot of iterative methods that can be written as algorithms (2).The main aim of this paper is to obtain a general study for the convergence, local and semilocal,

for these Newton-type of iterative methods given in (2).

296


2. Convergence Analysis for Two-Point Newton-Type Methods

In this section, we present both semilocal and local convergence analysis. In the first one,the hypotheses are imposed on the initial guess; and in the second, on the solution. The resultscan be applied for smooth and nonsmooth operators.

2.1. Local Convergence Analysis

We start the local analysis of method (2). Let v0 : [0,+∞)× [0,+∞)→ [0,+∞) be a nondecreasingcontinuous function. Assume that the equation

v0(t, t) = 1 (9)

has at least one positive root r0. Let also v : [0, r0)× [0, r0)→ [0,+∞) be a nondecreasing continuousfunction. Define function v on the interval [0, r0) by

v(t) =v(t, t)

1− v0(t, t)− 1. (10)

Assume that the equationv(t) = 0 (11)

has a minimal positive solution r. It follows that for each t ∈ [0, r)

0 ≤ v0(t, t) < 1 (12)

and0 ≤ v(t) < 1. (13)

Our analysis of method (2) will use the conditions (A):

• (a1) There exist a solution x∗ ∈ D of Equation (1), and B ∈ L(X, Y) such that B−1 ∈ L(Y, X).• (a2) Condition (9) holds and for each x, u ∈ D

‖B−1(L(x, u)− B)‖ ≤ v0(‖x− x∗‖, ‖u− y∗‖),

where v0 is defined previously, and r0 is given in (9).

Set D0 = D ∩ U(x∗, r0).• (a3) For each x, z ∈ D0, and any solution y ∈ D of Equation (1)

‖B−1(F1(x) + F2(x)− L(z, x)y)‖ ≤ v(‖z− y‖, ‖x− y‖)‖x− y‖,

where v is defined previously, and L(·, ·) : D0 × D0 → L(X, Y).• (a4) U(x∗, r) ⊂ D, where r is given in (10).• (a5)

ξ :=v(r, r)

1− v0(r, r)∈ [0, 1).

We are able to perform our local analysis of method (2) based on the aformentioned conditions (A).

Theorem 1. Assume that the conditions (A) hold. Then, sequence xk, defined by method (2) for x−1, x0 ∈U(x∗, r)− x∗, is well defined in U(x∗, r); remains in U(x∗, r); and converges to x∗. Finally, the followingestimates hold.

‖xk+1 − x∗‖ ≤ v(‖xk−1 − x∗‖, ‖xk − x∗‖)1− v0(‖xk−1 − x∗‖, ‖xk − x∗‖)‖xk − x∗‖ ≤ ‖xk − x∗‖ < r. (14)

297


The vector x∗ is the only solution of Equation (1) in U(x∗, r).

Proof. We will use mathematical induction on k.Let x, u ∈ U(x∗, r).Using (2), (a1) and (a2), we obtain

‖B−1(L(x, u)− B)‖ ≤ v0(‖x− x∗‖, ‖u− x∗‖) ≤ v0(r, r) < 1. (15)

Using the Banach lemma on invertible operators [20] and (15), we deduce that L(x, u)−1 ∈L(Y, X), and

‖L(x, u)−1B‖ ≤ 11− v0(‖x− x∗‖, ‖u− x∗‖) . (16)

In particular, estimate (16) holds for x = x0, so x1 is well defined by method (2) for k = 0.Using the definition of the method (2) (for k = 0); (a1), (a3), (13), and (16) (for k = 0) that

‖x1 − x∗‖ = ‖x0 − x∗ − L(x−1, x0)−1(x−1, x0)(F1(x0) + F2(x0))‖

= ‖[L(x−1, x0)−1B][B−1(F1(x0) + F2(x0)− L(x−1, x0)(x0 − x∗))]‖

≤ ‖L(x−1, x0)−1B‖‖B−1(F1(x0) + F2(x0)− L(x−1, x0)(x0 − x∗))‖

≤ v(‖x−1 − x∗‖, ‖x0 − x∗‖)1− v0(‖x−1 − x∗‖, ‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r, (17)

which shows estimate (14) for k = 0 and x1 ∈ U(x∗, r).Replace x0, x1 by xi, xi+1 in the preceding estimates to complete the induction for estimate (14).

Then, from the estimate

‖xi+1 − x∗‖ ≤ μ‖xi − x∗‖ < r, (18)

where

μ =v(‖x−1 − x‖, ‖x0 − x∗‖)

1− v0(‖x−1 − x∗‖, ‖x0 − x∗‖) ∈ [0, 1),

thus, limi→+∞ xi = x∗ and xi+1 ∈ U(x∗, r). Moreover, for the uniqueness part, let y∗ ∈ U(x∗, r) withF1(y∗) + F2(y∗) = 0. Using (a3), (a5), and estimate (17), we obtain in turn that

‖xi+1 − y∗‖ ≤ ‖L(xi−1, xi)−1B‖‖B−1(F1(xi) + F2(xi)− L(xi−1, xi)(xi − y∗))‖

≤ v(‖xi−1 − y∗‖, ‖xi − y∗‖)1− v0(‖xi−1 − y∗‖, ‖xi − y∗‖)‖xi − y∗‖

≤ μ‖xi − y∗‖ < μi+1‖x0 − y∗‖, (19)

which shows limi→+∞ xi = y∗—but, we showed limi→+∞ xi = x∗. Hence, we conclude that x∗ =

y∗.

Remark 1. • Condition (a3) can be replaced by the stronger: for each x, y ∈ D0

‖B−1(F1(x) + F2(x)− L(x)(x− y))‖ ≤ v1(‖x− y‖)‖x− y‖,

where function v1 is as v. However, for each t, t′ ≥ 0

v(t, t′) ≤ v1(t, t

′).

• Linear operator B does not necessarily depend on q, where q = x∗ or q = x0. It is used to determine theinvertibility of linear operator L(·, ·) appearing in the method. The invertibility of B can be assured by

298


an additional condition of the form ||I − B|| < 1 or in some other way. A possible choice for B is B = B(q)or B = F

′1(q).

• It follows from the definition of r0 and r that r0 ≥ r.

2.2. Semilocal Convergence Analysis

For the semilocal case, we also define some functions and parameters. Let w0 : [0,+∞) ×[0,+∞)→ [0,+∞) be a continuous and nondecreasing function.

Assume that the equationw0(t, t) = 1, (20)

has one smallest positive root that we denote by ρ0. Let w : [0, ρ0)× [0, ρ0)× [0, ρ0) → [0,+∞) bea nondecreasing continuous function. Moreover, for η, η ≥ 0, define parameters C1 and C2 by

C1 =w(η, η, 0)

1− w0(0, η),

C2 =w(0, η

1−C1, η)

1− w0(η, η1−C1

)

and function C : [0, ρ0)→ [0,+∞) by C(t) = w(t,t,t)1−w0(t,t)

. Assume that the equation

(C1C2

1− C(t)+ C1 + 1)η − t = 0 (21)

has one smallest positive root that we denote by ρ.The semilocal convergence analysis of method (2) will be based on conditions (H):

• (h1) There exists x−1, x0 ∈ D, and B ∈ L(X, Y) such that B−1 ∈ L(Y, X).• (h2) Condition (20) holds, and for each x ∈ D

‖B−1(L(z, x)− B)‖ ≤ w0(‖z− x0‖, ‖x− x0‖),

where w0 is defined previously and ρ0 is given in (20).

Set D1 = D⋂

U(x0, ρ0).• (h3) For L(·, ·) : D1 × D1 → L(X, Y), and each x, y, z ∈ D1

‖B−1(F1(y)− F1(x) + F2(y)− F2(x)− L(z, x)(y− x))‖≤ w(‖z− x0‖, ‖y− x0‖, ‖x− x0‖)‖y− x‖,

where w is defined previously.• (h4) U(x0, ρ) ⊆ D and condition (21) holds for ρ, where ‖x1 − x0‖ ≤ η and ‖x−1 − x0‖ ≤ η.

299


Then, using the hypotheses (H), we obtain the estimates:

‖x2 − x1‖ ≤ w(‖x−1 − x0‖, ‖x1 − x0‖, ‖x0 − x0‖)‖x1 − x0‖1− w0(‖x0 − x0‖, ‖x1 − x0‖) = C1‖x1 − x0‖,

‖x2 − x0‖ ≤ ‖x2 − x1‖+ ‖x1 − x0‖ ≤ (1 + C1)‖x1 − x0‖

=1− C2

11− C1

‖x1 − x0‖

<η

1− C1< ρ,

‖x3 − x2‖ ≤ w(‖x0 − x0‖, ‖x2 − x0‖, ‖x1 − x0‖)1− w0(‖x1 − x0‖, ‖x2 − x0‖) ‖x2 − x1‖

≤ w(0, η1−C1

, η)

1− w0(η, η1−C1

)‖x2 − x1‖ = C2‖x2 − x1‖,

‖x3 − x0‖ ≤ ‖x3 − x2‖+ ‖x2 − x1‖+ ‖x1 − x0‖≤ C2‖x2 − x1‖+ C1‖x1 − x0‖+ ‖x1 − x0‖≤ (C2C1 + C1 + 1)‖x1 − x0‖,

‖x4 − x3‖ ≤ w(‖x1 − x0‖, ‖x3 − x0‖, ‖x2 − x0‖)1− w0(‖x2 − x0‖, ‖x3 − x0‖) ‖x3 − x2‖

≤ C(ρ)‖x3 − x2‖ ≤ C(ρ)C2‖x2 − x1‖≤ C(ρ)C2C1‖x1 − x0‖,

(22)

similarly for i = 3, 4, . . .

‖xi+1 − xi‖ ≤ C(ρ)‖xi − xi−1‖ ≤ C(ρ)i−2‖x3 − x2‖,

‖xi+1 − x0‖ ≤ ‖xi+1 − xi‖+ ... + ‖x4 − x3‖+ ‖x3 − x0‖≤ C(ρ)‖xi − xi−1‖+ ... + C(ρ)‖x3 − x2‖

+(C2C1 + C1 + 1)‖x1 − x0‖≤ C(ρ)i−2‖x3 − x2‖+ ... + C(ρ)‖x3 − x2‖

+(C2C1 + C1 + 1)‖x1 − x0‖≤ (

1− C(ρ)i−1

1− C(ρ)C2C1 + C1 + 1)‖x1 − x0‖

< (C1C2

1− C(ρ)+ C1 + 1)η ≤ ρ, (23)

‖xi+j − xi‖ ≤ ‖xi+j − xi+j−1‖+ ‖xi+j−1 − xi+j−2‖+ ... + ‖xi+1 − xi‖≤ (C(ρ)i+j−3 + ... + C(ρ)i−2)‖x3 − x2‖≤ C(ρ)i−2 1− C(ρ)j−1

1− C(ρ)‖x3 − x2‖

≤ C(ρ)i−2 1− C(ρ)j−1

1− C(ρ)C2C1‖x1 − x0‖

≤ C(ρ)i−2 1− C(ρ)j−1

1− C(ρ)C2C1η. (24)

300


It follows from (23) that xi ∈ U(x0, ρ); and from (24) that sequence xi is complete in a Banachspace X. In particular, it converges to some x∗ ∈ U(x0, ρ). By letting i → +∞ in the estimate

‖B−1(F1(xi) + F2(xi))‖ = ‖B−1(F1(xi) + F2(xi)− F1(xi−1)− F2(xi−1)− Li−2,i−1(xi − xi−1))‖

≤ w(‖xi−2 − x0‖, ‖xi − x0‖, ‖xi−1 − x0‖)‖xi − xi−1‖1− w0(‖xi−1 − x0‖, ‖xi − x0‖) ≤ w(ρ, ρ, ρ)

1− w0(ρ, ρ)‖xi − xi−1‖,

we obtain F1(x∗) + F2(x∗) = 0. The uniqueness part is omitted as analogous to the one in the localconvergence case.

Hence, we can present our semilocal convergence result associated to the method (2).

Theorem 2. Assume that the conditions (H) hold. Then, sequence xk, defined by the method (2) for x−1, x0 ∈ D,is well defined in U(x0, ρ); remains in U(x0, ρ); and converges to a solution x∗ ∈ U(x0, ρ) of Equation (1).On the other hand, the vector x∗ is the only solution of Equation (1) in U(x0, ρ).

The same comments given in the previous remark hold.

3. Numerical Experiment

Consider the nondifferentiable system of equations⎧⎨⎩ 3x21x2 + x2

2 − 1 + |x1 − 1|3/2 = 0,

x41 + x1x3

2 − 1 + |x2|3/2 = 0.(25)

We therefore have an operator F : R2 → R2 such that F = F1 + F2, as in (1), with F1, F2 : R2 → R2,F1 = (F1

1, F12), F2 = (F2

1, F22), being

F11 (x1, x2) = 3x2

1x2 + x22 − 1 and F2

1 (x1, x2) = x41 + x1x3

2 − 1,

F12 (x1, x2) = |x1 − 1|3/2 and F2

2 (x1, x2) = |x2|3/2,

where the operator F1 is continuously Fréchet-differentiable and F2 is continuous but is a Fréchetnondifferentiable operator.

For u = (u1, uT2 ), v = (v1, v2)

T ∈ R2, we consider the divided difference of first order defined by[u, v; F] = ([u, v; F]ij)2

i,j=1 ∈ L(R2,R2), where

[u, v; F]i1 =

⎧⎨⎩Fi(u1, u2, . . . , vm)− Fi(u1, v2, . . . , vm)

u1 − v1, if u2 �= v2,

0, if u1 = v1,

[u, v; F]i2 =

⎧⎨⎩Fi(u1, u2, . . . , vm)− Fi(u1, v2, . . . , vm)

u2 − v2, if u2 �= v2,

0, if u2 = v2,

for i = 1, 2.The iterative processes given by (1) allow us to consider direct iterative processes, such as

(5) and (6); as well as iterative processes that use the decomposition method, such as (7) and (8).In this experiment, for the nondifferentiable system (25), we check that the application of the iterativeprocesses that use the decomposition method have better behavior than the direct methods.

301


To carry out this study, we will consider as an approximate solution of system (25):

x∗ = (0.9383410452297656, 0.3312445136375143) ,

the starting points x−1 = (5, 5) and x0 = (1, 0), and use a tolerance ‖xn+1 − xn‖ ≤ 10−16. In theseconditions, in Tables 1 and 2 we can see the results of the application of the direct iterative processes,the secant-type, and Kurchatov methods. Whereas in Tables 3 and 4 we can see the results of theapplication of the iterative processes that use the decomposition method, Newton-secant-type andNewton–Kuchatov methods. Observing the results obtained, it is evident that the best behavior of theiterative processes is given by (2) using the decomposition method.

Table 1. ‖x∗ − xn‖ for secant-type methods (5) and different values of the parameter λ.

n λ = 0 λ = 0.5 λ = 0.99

1 3.18484× 10−1 2.965× 10−1 4.54388× 10−2

2 5.21264× 10−2 4.13083× 10−2 3.74494× 10−3

3 3.66108× 10−3 2.35344× 10−3 2.94716× 10−5

4 2.59348× 10−4 7.2935× 10−5 3.69966× 10−9

5 1.30031× 10−6 1.24012× 10−7 2.10942× 10−15

6 4.42187× 10−10 6.07747× 10−12

7 1.11022× 10−15 1.11022× 10−16

Table 2. Kurchatov method (6).

n ‖x∗ − xn‖1 2.99754× 10−1

2 1.07269× 10−1

3 4.20963× 10−2

4 8.37098× 10−3

5 2.78931× 10−4

6 9.0784× 10−8

7 2.9826× 10−11

Table 3. ‖x∗ − xn‖ for Newton-secant-type methods (7) and different values of the parameter λ.

n λ = 0 λ = 0.5 λ = 0.99

1 2.3538× 10−1 1.00278× 10−1 4.29554× 10−2

2 3.48717× 10−1 2.88094× 10−2 2.53626× 10−3

3 1.47537× 10−1 1.90518× 10−3 9.06208× 10−6

4 3.4371× 10−2 8.39107× 10−6 1.9925× 10−10

5 3.08399× 10−3 4.78016× 10−9

6 4.63665× 10−5 7.38298× 10−15

7 4.05776× 10−8

8 2.96929× 10−13

Table 4. Newton–Kurchatov method (8).

n ‖x∗ − xn‖1 6.1659× 10−1

2 4.75269× 10−3

3 6.29174× 10−5

4 3.37027× 10−9

302


Remark 2. In the above example, we have selected the initial guess in a region where the operator is not smooth.The methods can be applied to systems where the operator is not smooth at the solution.

For instance, for the system: { |x21 − 1|+ x2 − 1 = 0,

x1 + x22 − 1 = 0,

(26)

the solution is (1, 1). If we take as initial guess (0.5, 0.5), the Steffensen method gives as errors 3.12× 10−2,5.48× 10−4, 1.92× 10−7, 2.15× 10−14, we observe its second order.

4. Boundary Value Problem: Discretization via the Multiple Shooting Method

We will use the multiple shooting method for the discretization of boundary problems of the type

y′′(t) = f (t, y(t), y′(t)), y(a) = α, y(b) = β. (27)

Thus, we should find the solution of the following nonlinear system of equations F(s) = 0, whereF : RN −→ RN and⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

F1(s0, s1, . . . , sN−1) = s1 − y′(t1; s0)

F2(s0, s1, . . . , sN−1) = s2 − y′(t2; s0, s1)...

FN−1(s0, s1, . . . , sN−1) = sN−1 − y′(tN−1; s0, s1, . . . , sN−2)

FN(s0, s1, . . . , sN−1) = β− y(tN ; s0, s1, sN−2, sN−1).

for a discretization of [a, b] with N subintervals,

TN j, T = b− a, j = 0, 1, . . . , N.

We consider the secant-type method⎧⎪⎪⎨⎪⎪⎩Given y−1, y0 ∈ D,

zn = λnyn + (1− λn)yn−1, λn ∈ [0, 1),

yn+1 = yn − [zn, yn; F]−1F(yn), n ≥ 0,

(28)

where λn is such that ||zn − xn|| ≤ Tol for a given tolerance, and Newton’s method⎧⎨⎩ Given y0 ∈ D,

yn+1 = yn − F(yn)−1F(yn), n ≥ 0.

(29)

We perform a numerical comparison between both methods. As we can see, in the multipleshooting method, the iterative schemes are used as black boxes.

303


For the initial slope�s0 =(s0

0, s01, . . . , s0

N−1), we propose⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

s00 =

β− α

b− a=

y(tN)− y(t0)

tN − t0,

s01 =

y(tN)− y(t1; s0)

tN − t1,

s02 =

y(tN)− y(t2; s0, s1)

tN − t2,

...

s0N−1 =

y(tN)− y(tN−1; s0, s1, . . . , sN−2)

tN − tN−1.

We analyze this particular example ([21], p. 554):

y′′(t) = τ · sinh(τ · y(t)),

y(0) = 0, y(1) = 1.

We take τ = 2.5 and N = 4 subintervals.This problem admits the solution:

y(t) =2τ

arg sinh(

s2· sn (τt, 1− s2/4)

cn (τt, 1− s2/4)

),

wheres = y′(0) = 0.3713363932677645

and sn(·, ·) and cn(·, ·) are the Jacobi elliptic functions.

Newton’s method (29),

n ‖F(�sn)‖∞ ‖y(t)− yn‖∞ ‖y′(t)− y′n‖∞

0 100 10−1 100

1 10−1 10−1 10−1

2 10−2 10−2 10−2

3 10−4 10−4 10−4

4 10−7 10−7 10−7

5 10−15 10−15 10−14

Secant-type method (28),

n ‖F(�sn)‖∞ ‖y(t)− yn‖∞ ‖y′(t)− y′n‖∞ ‖F′(yn)− [yn, xn; F]‖∞

0 100 10−1 100 10−6

1 10−1 10−1 10−1 10−6

2 10−2 10−2 10−2 10−7

3 10−4 10−4 10−4 10−6

4 10−7 10−7 10−7 10−6

5 10−15 10−15 10−14 10−6

304


The methods using Jacobians obtain their order of convergence. However, in this example, thecomputation of the derivatives involves the approximation of a more complicated problem. For thisreason, the methods free of derivatives are preferred, see [21]. Of course, we need to compute a goodapproximation to the Jacobian, this is the motivation of our parameters λn. For more similar examplesand conclusions, we refer [22].

Remark 3. In many cases, when we manipulate an image, some random noise appears. This noise makes thelater steps of processing the image difficult and inaccurate.

Let f : Ω → R be a noise signal or image.Introducing the variable w:

w =∇u√|∇u|2 ,

the Total-Variation model is equivalent to the nonlinear and nondifferentiable system:

−∇ · w + λ(u− f ) = 0,

w√|∇u|2 −∇u = 0.

This system should be discretized using finite differences and the associated nonlinear system of equationscan be approximated by our family (see [23] for more details).

5. Conclusions

This paper was devoted to the analysis of a general family of two-point Newton-type methodsfor smooth and nonsmooth operators. We have considered two types of convergence—semilocaland local. The family includes a great number of methods. We have applied the schemes to severalinteresting problems, in particular to nonsmooth nonlinear systems, boundary problems, and imagedenoising models.

Author Contributions: All the authors have been contributed similarly and they are participated in all the work.


Acknowledgments: Research of the first and third authors supported in part by Programa de Apoyo a lainvestigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18and by MTM2015-64382-P. Research of the fourth and fifth authors supported by Ministerio de Economía yCompetitividad under grant MTM2014-52016-C2-1P.


References

1. Amat, S.; Busquier, S. A modified secant method for semismooth equations. Appl. Math. Lett. 2003, 16,877–881. [CrossRef]

2. Amat, S.; Busquier, S. On a higher order secant method. Appl. Math. Comput. 2003, 141, 321–329. [CrossRef]3. Amat, S.; Busquier, S. A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal.

Appl. 2006, 324, 1084–1092. [CrossRef]4. Argyros, I.K.; Magreñán, A.A. Iterative Methods and Their Dynamics with Applications: A Contemporary Study;

CRC Press: Boca Raton, FL, USA, 2017.5. Argyros, I.K.; Magreñán, A.A. A Contemporary Study of Iterative Methods; Academic Press: Cambridge, MA,

USA, 2018.6. Grau-Sánchez, M. Noguera, M.; Amat, S. On the approximation of derivatives using divided difference

operators preserving the local convergence order of iterative methods. J. Comput. Appl. Math. 2013, 237,363–372.

7. Amat, S.; Bermúdez, C.; Busquier, S.; Mestiri, D. A family of Halley-Chebyshev iterative schemes fornon-Fréchet differentiable operators. J. Comput. Appl. Math. 2009, 228, 486-493. [CrossRef]

305


8. Barton, M. Solving polynomial systems using no-root elimination blending schemes. Comput.-Aided Des.2011, 43, 1870–1878. [CrossRef]

9. Barton, M.; Elber, G.; Hanniel, I. Topologically guaranteed univariate solutions of underconstrainedpolynomial systems via no-loop and single-component tests. Comput.-Aided Des. 2011, 43, 1035–1044.[CrossRef]

10. Hanniel, I.; Elber, G. Subdivision termination criteria in subdivision multivariate solvers using dualhyperplanes representations. Comput.-Aided Des. 2007, 39, 369–378. [CrossRef]

11. Chen, J.; Shen, Z. Convergence analysis of the secant type methods. Appl. Math. Comput. 2007, 188, 514–524.[CrossRef]

12. Hongmin, R.; Qingiao, W. The convergence ball of the Secant method under Hölder continuous divideddifferences. J. Comput. Appl. Math. 2006, 194, 284–293. [CrossRef]

13. Kewei, L. Homocentric convergence ball of the Secant method. Appl. Math. J. Chin. Univ. Ser. B 2007, 22,353–365.

14. Ren, H.; Argyros, I.K. Local convergence of efficient Secant-type methods for solving nonlinear equations.Appl. Math. Comput. 2012, 218, 7655–7664. [CrossRef]

15. Zincenko, A.I. Some Approximate Methods of Solving Equations with Non-Differentiable Operators; DopovidiAkad Nauk: 1963; pp. 156–161.

16. Hernández, M.A.; Rubio, M.J. A uniparametric family of iterative processes for solving nondifferentiableequations. J. Math. Anal. Appl. 2002, 275, 821–834. [CrossRef]

17. Hernández, M.A.; Rubio, M.J.; Ezquerro, J.A. Secant-like methods for solving nonlinear integral equations ofthe Hammerstein type. J. Comput. Appl. Math. 2000, 115, 245–254. [CrossRef]

18. Kurchatov, V.A. On a method of linear interpolation for the solution of funcional equations. Dolk. Akad. NaukSSSR 1971, 198, 524–526; translation in Soviet Math. Dolk. 1971, 12, 835–838. (In Russian)

19. Hernández, M.A.; Rubio, M.J. On a Newton-Kurchatov-type Iterative Process. Numer. Funct. Anal. 2016, 37,65–79. [CrossRef]

20. Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Pergamon Press: Oxford, UK, 1982.21. Stoer, J.; Bulirsch, R. Introduction to Numerical Analysis, 2nd ed.; Springer: New York, NY, USA, 1993.22. Alarcón, V.; Amat, S.; Busquier, S.; López, D.J. A Steffensen’s type method in Banach spaces with applications

on boundary value problems. J. Comput. Appl. Math. 2008, 216, 243–250. [CrossRef]23. Amat, S.; Argyros, I.K.; Busquier, S.; Hernández-Verón, M.A.; Martínez, E. A unified convergence analysis

for single step-type methods for non-smooth operators. 2019, submitted.


306

mathematics

Article

A Modified Fletcher–Reeves Conjugate GradientMethod for Monotone Nonlinear Equations withSome Applications

Auwal Bala Abubakar 1,2 , Poom Kumam 1,3,4,* , Hassan Mohammad 2 ,

Aliyu Muhammed Awwal 1,5 and Kanokwan Sitthithakerngkiet 6

1 KMUTTFixed Point Research Laboratory, SCL 802 Fixed Point Laboratory, Science Laboratory Building,Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi(KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

2 Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano 700241, Nigeria3 Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building,

King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod,Thrung Khru, Bangkok 10140, Thailand


5 Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria6 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North

Bangkok, 1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800, Thailand* Correspondence: [email protected]

Received: 24 June 2019; Accepted: 5 August 2019; Published: 15 August 2019

Abstract: One of the fastest growing and efficient methods for solving the unconstrainedminimization problem is the conjugate gradient method (CG). Recently, considerable efforts havebeen made to extend the CG method for solving monotone nonlinear equations. In this researcharticle, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection methodfor constrained monotone nonlinear equations. The method possesses sufficient descent propertyand its global convergence was proved using some appropriate assumptions. Two sets of numericalexperiments were carried out to show the good performance of the proposed method compared withsome existing ones. The first experiment was for solving monotone constrained nonlinear equationsusing some benchmark test problem while the second experiment was applying the method in signaland image recovery problems arising from compressive sensing.

Keywords: nonlinear equations; conjugate gradient method; projection method; convex constraints;signal and image processing

MSC: 65K05; 90C52; 90C56; 94A08

1. Introduction

In this paper, we are considering a system of nonlinear monotone equations of the form

F(x) = 0, subject to x ∈ E, (1)

where E ⊆ Rn is closed and convex, F : Rn → Rm, (m ≥ n) is continuous and monotone, which means

〈F(x)− F(y), (x− y)〉 ≥ 0, ∀x, y ∈ Rn.



A well-known fact is that under the above assumption, the solution set of (1) is convex unlessis empty. It is important to mention that nonlinear monotone equations arise in many practicalapplications. These and other reasons motivate researchers to develop a large number of classof Iterative methods for solving such systems, for example, see [1–7] among others. In addition,convex constrained equations have application in many scientific fields, some of which are theeconomic equilibrium problems [8], the chemical equilibrium systems [9], etc. Several algorithmswere developed to solve (1), among them, are the trust-region [10] and the Levenberg-Marquardtmethod [11]. Moreover, the requirement to compute and store the matrix in every iteration makesthem ineffective for large-scale nonlinear equations.

Conjugate gradient (CG) methods are efficient for solving large-scale optimization and nonlinearsystems because of their low memory requirements. This forms part of the reason several Iterativemethods with CG-like directions are proposed in recent years [12,13]. Initially, CG methods andtheir modified versions are proposed for unconstrained optimization problems [14–19]. Inspired bythem, in the last decade, many authors used the CG direction to solve nonlinear monotone equationsfor both constrained and unconstrained cases. Since in this article, we are interested in solvingnonlinear monotone equations with convex constraints, we will only discuss existing methods withsuch properties.

Many methods for solving nonlinear monotone equations with convex constraints have beenpresented in the last decade. For examples, Xiao and Zhu [20] presented a CG method, whichcombines the well-known CG-DESCENT method in [17] and the projection method by Solodovand Svaiter [21]. Liu et al. [22] proposed two CG methods with projection strategy for solving (1).In [23], a modification of the method in [20] was presented by Liu and Li. One of the reasons for themodification was to improve the numerical performance of the method in [20]. Also, Sun and Liu [24]presented derivative-free projection methods for solving nonlinear equations with convex constraints.These methods are the combination of some existing CG methods and the well-known projectionmethod. In addition, a hybrid CG projection method for convex constrained equations was developedin [25]. Ou and Li [26] proposed a combination of a scaled CG method and the projection strategy tosolve (1). Furthermore, Ding et al. [27] extended the Dai and Kou (DK) CG method to solve (1) byalso combining it with the projection method. Just recently, to popularize the Dai-Yuan (DY) method,Liu and Feng [28] proposed a modified DY method for solving convex constraints monotone equation.The global convergence was also obtained under certain assumptions and finally, some numericalresults were reported to show its efficiency.

Inspired by some the above proposals, we present a simple modification of the Fletcher–Reeves(FR) conjugate gradient method [19] considered in [12] to solve nonlinear monotone equations withconvex constraints. The modification ensures that the direction is automatically descent, improves itsnumerical performance and still inherits the nice convergence properties of the method. Under suitableassumptions, we establish the global convergence of the proposed algorithm. Numerical experimentspresented show the good performance and competitiveness of the method. In addition, the proposedmethod has the advantages of the direct methods [29] such as boundary control method by Belishevand Kuryiev [30], the globally convergent method proposed by Beilina and Klibanov [31] and methodbased on the multidimensional analogs of Gelfand–Levitan–Krein equations [32,33]. The proposedmethod can be seen as a local method that looks for the closest root. However, there are several globalnonlinear solvers that guarantee finding all roots inside a domain and within a very fine double-floataccuracy. In some cases a combination of subdivision-based polynomial solver with a decompositionalgorithm are employed in order to handle large and complex systems (see for examples [34–36] andreferences therein).

The remaining part of this article is organized as follows. In Section 2, we mention somepreliminaries and present the proposed method. The global convergence of the method is establishedin Section 3. Finally, Section 4 reports some numerical results to show the performance of the method

308


in solving monotone nonlinear equations with convex constraints, and also apply it to recover a noisysignal and a blurred image.

2. Algorithm

In this section, we define the projection map together with its well-known properties, give someuseful assumptions and finally present the proposed algorithm. Throughout this article, ‖ · ‖ denotesthe Euclidean norm.

Definition 1. Let E ⊂ Rn be nonempty closed and convex set. Then for any x ∈ Rn, its projection onto E isdefined as

PE(x) = arg min{‖x− y‖ : y ∈ E.}

The following lemma gives some properties of the projection map.

Lemma 1 ([37]). Suppose E ⊂ Rn is nonempty, closed and convex set. Then the following statements are true:

1. 〈x− PE(x), PE(x)− z〉 ≥ 0, ∀x, z ∈ Rn.2. ‖PE(x)− PE(y)‖ ≤ ‖x− y‖, ∀x, y ∈ Rn.3. ‖PE(x)− z‖2 ≤ ‖x− z‖2 − ‖x− PE(x)‖2, ∀x, z ∈ Rn.

Throughout, we suppose the followings

(C1) The solution set of (1), denoted by E′, is nonempty.

(C2) The mapping F is monotone.(C3) The mapping F is Lipschitz continuous, that is there exists a positive constant L such that

‖F(x)− F(y)‖ ≤ L‖x− y‖, ∀x, y ∈ Rn.

Our algorithm is motivated by the work of Papp and Rapajic in [12]. In the paper, theymodified the well known Fletcher–Reeves conjugate gradient method to solve unconstrainednonlinear monotone equation. The modification was adding the term −θkF(xk) to the directionof Fletcher–Reeves. The parameter θk was then determined in three different ways and three differentdirections were proposed, namely, M3TFR1, M3TFR2 and M3TFR3. The direction we are interested inis M3TFR1 and is defined as:

dk =

{−F(xk), if k = 0,

−F(xk) + βFRk wk−1 + θkF(xk), if k ≥ 1,

(2)

where,

βFRk =

‖F(xk)‖2

‖F(xk−1)‖2 , θk = − F(xk)Twk−1

‖F(xk−1)‖2 , wk−1 = zk−1 − xk−1, zk−1 = xk−1 + αk−1dk−1.

It follows thatF(xk)

Tdk = −‖F(xk)‖2.

Using same modification proposed in [3], we modify the direction (2) as follows

dk =

⎧⎨⎩−F(xk), if k = 0,

−F(xk) +‖F(xk)‖2wk−1−F(xk)

Twk−1F(xk)max{μ‖wk−1‖‖F(xk)‖,‖F(xk−1)‖2} , if k ≥ 1,

(3)

where μ > 0 is a positive constant. The difference between the M3TFR1 direction and thedirection proposed in this paper is the scaling term appearing in the denominator of Equation (3)

309


i.e., max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}. This modification was shown to have a very good numericalperformance in [3] and also helps in obtaining the boundedness of the direction easily.

Remark 1. Note the the parameter μ is chosen to be strictly positive because if μ ≤ 0 then

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2} = ‖F(xk−1)‖2.

This means that the direction dk will always be M3TFR1 given by (2).


To prove the global convergence of Algorithm 1, the following results are needed.

Algorithm 1: A modified descent Fletcher–Reeves CG method (MFRM).

Step 0. Select the initial point x0 ∈ Rn, parameters μ > 0, σ > 0, 0 < ρ < 1, Tol > 0, and setk := 0.

Step 1. If ‖F(xk)‖ ≤ Tol, stop, otherwise go to Step 2.Step 2. Find dk using (3).Step 3. Find the step length αk = γρmk where mk is the smallest non-negative integer m suchthat

− 〈F(xk + αkdk), dk〉 ≥ σαk‖F(xk + αkdk)‖‖dk‖2. (4)

Step 4. Set zk = xk + αkdk. If zk ∈ E and ‖F(zk)‖ ≤ Tol, stop. Else compute

xk+1 = PE[xk − ζkF(zk)]

where

ζk =F(zk)

T(xk − zk)

‖F(zk)‖2 .

Step 5. Let k = k + 1 and go to Step 1.

Lemma 2. Let dk be defined by Equation (3), then

dTk F(xk) = −‖F(xk)‖2 (5)

and

‖F(xk)‖ ≤ ‖dk‖ ≤(

1 +2μ

)‖F(xk)‖. (6)

Proof. By Equation (3), suppose k = 0,

dTk F(xk) = −F(xk)

T F(xk) = −‖F(xk)‖2.

Now suppose k > 0,

dTk F(xk) = −F(xk)

T F(xk) +(‖F(xk)‖2wk−1)

T F(xk)− (F(xk)Twk−1F(xk))

T F(xk)

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}

= −‖F(xk)‖2 +‖F(xk)‖2wT

k−1F(xk)− F(xk)T(wT

k−1F(xk))F(xk)

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}

= −‖F(xk)‖2 +‖F(xk)‖2wT

k−1F(xk)− ‖F(xk)‖2wTk−1F(xk)

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}= −‖F(xk)‖2.

(7)

310


Using Cauchy–Schwartz inequality, we get

‖F(xk)‖ ≤ ‖dk‖. (8)

Furthermore, since max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2} ≥ μ‖wk−1‖‖F(xk)‖, then,

‖dk‖ =∥∥∥∥−F(xk) +

‖F(xk)‖2wk−1 − (F(xk)Twk−1)F(xk)

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}∥∥∥∥

≤ ‖− F(xk)‖+ ‖‖F(xk)‖2wk−1 − (F(xk)Twk−1)F(xk)‖

max{μ‖wk−1‖‖F(xk)‖, ‖F(xk−1)‖2}≤ ‖F(xk)‖+ ‖F(xk)‖2‖wk−1‖

μ‖wk−1‖‖F(xk)‖ +‖F(xk)

Twk−1F(xk)‖μ‖wk−1‖‖F(xk)‖

≤ ‖F(xk)‖+ ‖F(xk)‖2‖wk−1‖μ‖wk−1‖‖F(xk)‖ +

‖F(xk)‖2‖wk−1‖μ‖wk−1‖‖F(xk)‖

= ‖F(xk)‖+ 2‖F(xk)‖μ

=

(1 +

2μ

)‖F(xk)‖.

(9)

Combining (8) and (9), we get the desired result.

Lemma 3. Suppose that assumptions (C1)–(C3) hold and the sequences {xk} and {zk} are generated byAlgorithm 1. Then we have

αk ≥ ρ min

{1,

‖F(xk)‖2

(L + σ)‖F(xk +αkρ dk)‖‖dk‖2

}

Proof. Suppose αk �= ρ, then αkρ does not satisfy Equation (4), that is

− F(

xk +αkρ

dk

)< σ

αkρ‖F(xk +

αkρ

dk)‖‖dk‖2.

This combined with (7) and the fact that F is Lipschitz continuous yields

‖F(xk)‖2 = −F(xk)Tdk

=

(F(xk +

αkρ

dk)− F(xk)

)Tdk − FT

(xk +

αkρ

dk

)dk

≤ Lαkρ‖F(xk +

αkρ

dk)‖‖dk‖2 + σαkρ‖F(xk +

αkρ

dk)‖‖dk‖2

=L + σ

ραk‖F(xk +

αkρ

dk)‖‖dk‖2.

(10)

The above equation implies

αk ≥ ρ min‖F(xk)‖2

(L + σ)‖F(xk +αkρ dk)‖‖dk‖2 ,

which completes the proof.

311


Lemma 4. Suppose that assumptions (C1)–(C3) holds, then the sequences {xk} and {zk} generated byAlgorithm 1 are bounded. Moreover, we have

limk→∞

‖xk − zk‖ = 0 (11)

andlimk→∞

‖xk+1 − xk‖ = 0. (12)

Proof. We will start by showing that the sequences {xk} and {zk} are bounded. Suppose x ∈ E′,

then by monotonicity of F, we get

〈F(zk), xk − x〉 ≥ 〈F(zk), xk − zk〉. (13)

Also by definition of zk and the line search (4), we have

〈F(zk), xk − zk〉 ≥ σα2k‖F(zk)‖‖dk‖2 ≥ 0. (14)

So, we have

‖xk+1 − x‖2 = ‖PE[xk − ζkF(zk)]− x‖2 ≤ ‖xk − ζkF(zk)− x‖2

= ‖xk − x‖2 − 2ζ〈F(zk), xk − x〉+ ‖ζF(zk)‖2

≤ ‖xk − x‖2 − 2ζk〈F(zk), xk − zk〉+ ‖ζF(zk)‖2

= ‖xk − x‖2 −( 〈F(zk), xk − zk〉

‖F(zk)‖)2

≤ ‖xk − x‖2.

(15)

Thus the sequence {‖xk − x‖} is non increasing and convergent, and hence {xk} is bounded.Furthermore, from Equation (15), we have

‖xk+1 − x‖2 ≤ ‖xk − x‖2, (16)

and we can deduce recursively that

‖xk − x‖2 ≤ ‖x0 − x‖2, ∀k ≥ 0.

Then from assumption (C3), we obtain

‖F(xk)‖ = ‖F(xk)− F(x)‖ ≤ L‖xk − x‖ ≤ L‖x0 − x‖.

If we let L‖x0 − x‖ = κ, then the sequence {F(xk)} is bounded, that is,

‖F(xk)‖ ≤ κ, ∀k ≥ 0. (17)

By the definition of zk, Equation (14), monotonicity of F and the Cauchy–Schwatz inequality,we get

σ‖xk − zk‖ = σ‖αkdk‖2

‖xk − zk‖ ≤〈F(zk), xk − zk〉‖xk − zk‖ ≤ 〈F(xk), xk − zk〉

‖xk − zk‖ ≤ ‖F(xk)‖. (18)

312


The boundedness of the sequence {xk} together with Equations (17) and (18), implies the sequence{zk} is bounded.

Now, as {zk} is bounded, then for any x ∈ E′, the sequence {zk − x} is also bounded, that is,

there exists a positive constant ν > 0 such that

‖zk − x‖ ≤ ν.

This together with assumption (C3), this yields

‖F(zk)‖ = ‖F(zk)− F(x)‖ ≤ L‖zk − x‖ ≤ Lν.

Therefore, using Equation (15), we have

σ2

(Lν)2 ‖xk − zk‖4 ≤ ‖xk − x‖2 − ‖xk+1 − x‖2,

which implies

σ2

(Lν)2

∞

∑k=0‖xk − zk‖4 ≤

∞

∑k=0

(‖xk − x‖2 − ‖xk+1 − x‖2) ≤ ‖x0 − x‖ < ∞. (19)

Equation (19) implieslimk→∞

‖xk − zk‖ = 0.

However, using statement 2 of Lemma 1, the definition of ζk and the Cauchy-Schwartz inequality,we have

‖xk+1 − xk‖ = ‖PE[xk − ζkF(zk)]− xk‖

≤ ‖xk − ζkF(zk)− xk‖

= ‖ζkF(zk)‖

= ‖xk − zk‖,

(20)

which yieldslimk→∞

‖xk+1 − xk‖ = 0.

Remark 2. By Equation (11) and definition of zk, then

limk→∞

αk‖dk‖ = 0. (21)

Theorem 1. Suppose that assumption (C1)–(C3) holds and let the sequence {xk} be generated byAlgorithm 1, then

lim infk→∞

‖F(xk)‖ = 0. (22)

Proof. Assume that Equation (22) is not true, then there exists a constant ε > 0 such that

‖F(xk)‖ ≥ ε, ∀k ≥ 0. (23)

313


Combining (8) and (23), we have

‖dk‖ ≥ ‖F(xk)‖ ≥ ε, ∀k ≥ 0.

As wk = xk + αkdk and limk→∞ ‖xk − zk‖ = 0, we get limk→∞ αk‖dk‖ = 0 and

limk→∞

αk = 0. (24)

On the other side, if M =(

1 + 2μ

)κ, Lemma 3 and Equation (9) implies αk‖dk‖ ≥ ρ ε2

(L+σ)MLν,

which contradicts with (24). Therefore, (22) must hold.


To test the performance of the proposed method, we compare it with accelerated conjugategradient descent (ACGD) and projected Dai-Yuan (PDY) methods in [27,28], respectively. In addition,MFRM method is applied to solve signal and image recovery problems arising in compressive sensing.All codes were written in MATLAB R2018b and run on a PC with intel COREi5 processor with 4GB ofRAM and CPU 2.3GHZ. All runs were stopped whenever ‖F(xk)‖ < 10−5. The parameters chosen foreach method are as follows:

MFRM method: γ = 1, ρ = 0.9, μ = 0.01, σ = 0.0001.ACGD method: all parameters are chosen as in [27].PDY method: all parameters are chosen as in [28].

We tested eight problems with dimensions of n = 1000, 5000, 10,000, 50,000, 100,000 and 6initial points: x1 = (0.1, 0.1, · · · , 1)T , x2 = (0.2, 0.2, · · · , 0.2)T , x3 = (0.5, 0.5, · · · , 0.5)T , x4 =

(1.2, 1.2, · · · , 1.2)T , x5 = (1.5, 1.5, · · · , 1.5)T , x6 = (2, 2, · · · , 2)T . In Tables 1–8, the number ofIterations (Iter), number of function evaluations (Fval), CPU time in seconds (time) and the norm at theapproximate solution (NORM) were reported. The symbol ‘−’ is used when the number of Iterationsexceeds 1000 and/or the number of function evaluations exceeds 2000.

The test problems are listed below, where the function F is taken as F(x) =

( f1(x), f2(x), . . . , fn(x))T .Problem 1 [38] Exponential Function.

f1(x) = ex1 − 1,

fi(x) = exi + xi − 1, for i = 2, 3, ..., n,

and E = Rn+.

Problem 2 [38] Modified Logarithmic Function.

fi(x) = ln(xi + 1)− xin

, for i = 2, 3, ..., n,

and E = {x ∈ Rn :

n

∑i=1

xi ≤ n, xi > −1, i = 1, 2, . . . , n}.

Problem 3 [6] Nonsmooth Function.

fi(x) = 2xi − sin |xi|, i = 1, 2, 3, ..., n,

and E = {x ∈ Rn :

n

∑i=1

xi ≤ n, xi ≥ 0, i = 1, 2, . . . , n}.

It is clear that problem 3 is nonsmooth at x = 0.

314


Problem 4 [38] Strictly Convex Function I.

fi(x) = exi − 1, for i = 1, 2, ..., n,

and E = Rn+.

Problem 5 [38] Strictly Convex Function II.

fi(x) =in

exi − 1, for i = 1, 2, ..., n,

and E = Rn+.

Problem 6 [39] Tridiagonal Exponential Function

f1(x) = x1 − ecos(h(x1+x2)),

fi(x) = xi − ecos(h(xi−1+xi+xi+1)), for i = 2, ..., n− 1,

fn(x) = xn − ecos(h(xn−1+xn)),

h =1

n + 1and E = R

n+.

Problem 7 [40] Nonsmooth Function

fi(x) = xi − sin |xi − 1|, i = 1, 2, 3, ..., n.

and E = {x ∈ Rn :

n

∑i=1

xi ≤ n, xi ≥ −1, i = 1, 2, . . . , n}.

Problem 8 [27] Penalty 1

ti =n

∑i=1

x2i , c = 10−5

fi(x) = 2c(xi − 1) + 4(ti − 0.25)xi, i = 1, 2, 3, ..., n.

and E = Rn+.

To show in detail the efficiency and robustness of all methods, we employ the performanceprofile developed in [41], which is a helpful process of standardizing the comparison of methods.Suppose that we have ns solvers and nl problems and we are interested in using either number ofIterations, CPU time or number of function evaluations as our measure of performance; so we let kl,sto be the number of iterations, CPU time or number of function evaluations required to solve problemby solver s. To compare the performance on problem l by a solver s with the best performance by anyother solver on this problem, we use the performance ratio rl,s defined as

rl,s =kl,s

min{kl,s : s ∈ S} ,

where S is the set of solvers.The overall performance of the solver is obtained using the (cumulative) distribution function for

the performance ratio P. So if we let

P(t) =1nl

size{l ∈ L : rl,s ≤ t},

315


then P(t) is the probability for solver s ∈ S that a performance ratio rl,s is within a factor t ∈ R of thebest possible ratio. If the set of problems L is large enough, then the solvers with the large probabilityP(t) are considered as the best.

Table 1. Numerical results for modified Fletcher–Reeves (MFRM), accelerated conjugate gradientdescent (ACGD) and projected Dai-Yuan (PDY) for problem 1 with given initial points and dimensions.

MFRM ACGD PDY

Dimension Initial Point Iter Fval Time Norm Iter Fval Time Norm Iter Fval Time Norm

1000

x1 23 98 0.42639 9.01 × 10−6 8 34 0.21556 9.26 × 10−6 12 49 0.19349 9.18 × 10−6

x2 7 35 0.019885 8.82 × 10−6 9 39 0.086582 3.01 × 10−6 13 53 0.07318 6.35 × 10−6

x3 8 40 0.011238 9.74 × 10−6 9 38 0.034359 4.02 × 10−6 14 57 0.01405 5.59 × 10−6

x4 15 70 0.066659 6.01 × 10−6 16 67 0.017188 9.22 × 10−6 15 61 0.01421 4.07 × 10−6

x5 5 31 0.16103 0 18 75 0.11646 4.46 × 10−6 14 57 0.08690 9.91 × 10−6

x6 31 134 0.03232 7.65 × 10−6 25 104 0.042967 6.74 × 10−6 40 162 0.04060 9.70 × 10−6

5000

x1 8 38 0.053865 5.63 × 10−6 9 38 0.023729 3.89 × 10−6 13 53 0.02775 6.87 × 10−6

x2 8 40 0.036653 2.59 × 10−6 9 38 0.021951 6.65 × 10−6 14 57 0.02974 4.62 × 10−6

x3 8 40 0.030089 6.41 × 10−6 9 39 0.019317 8.01 × 10−6 15 61 0.04353 4.18 × 10−6

x4 16 74 0.081741 4.71 × 10−6 17 71 0.05235 8.12 × 10−6 15 61 0.03288 9.08 × 10−6

x5 5 31 0.030748 0 18 75 0.038894 8.14 × 10−6 15 61 0.03556 7.30 × 10−6

x6 31 134 0.087531 8.1 × 10−6 26 108 0.053473 7.96 × 10−6 39 158 0.10419 9.86 × 10−6

10,000

x1 5 26 0.03829 3.7 × 10−6 9 39 0.044961 5.5 × 10−6 13 53 0.05544 9.70 × 10−6

x2 8 40 0.055099 3.64 × 10−6 9 39 0.0358 9.39 × 10−6 14 57 0.06201 6.53 × 10−6

x3 8 40 0.049974 5.44 × 10−6 10 43 0.04176 2.12 × 10−6 15 61 0.08704 5.90 × 10−6

x4 16 74 0.125 6.61 × 10−6 18 75 0.066316 4.58 × 10−6 16 65 0.07797 4.28 × 10−6

x5 5 31 0.048751 0 18 75 0.11807 7.86 × 10−6 39 158 0.20751 7.97 × 10−6

x6 28 122 0.13649 7.18 × 10−6 27 112 0.10593 6.22 × 10−6 87 351 0.36678 9.93 × 10−6

50,000

x1 5 26 0.1584 3.58 × 10−6 10 43 0.15918 2.33 × 10−6 14 57 0.23129 7.12 × 10−6

x2 8 40 0.18044 8.1 × 10−6 10 43 0.16252 3.97 × 10−6 15 61 0.23975 4.91 × 10−6

x3 8 40 0.186 4.54 × 10−6 10 43 0.15707 4.67 × 10−6 16 65 0.24735 4.37 × 10−6

x4 17 78 0.31567 5.47 × 10−6 19 79 0.27474 4.1 × 10−6 38 154 0.55277 7.54 × 10−6

x5 5 31 0.18586 0 18 75 0.27118 5.06 × 10−6 177 712 2.29950 9.44 × 10−6

x6 20 90 0.39237 6.44 × 10−6 28 116 0.35197 7.69 × 10−6 361 1449 4.63780 9.74 × 10−6

100,000

x1 5 26 0.26116 4.59 × 10−6 10 42 0.28038 3.29 × 10−6 15 61 0.50090 3.39 × 10−6

x2 9 43 0.35288 1.59 × 10−6 10 42 0.28999 5.62 × 10−6 15 61 0.45876 6.94 × 10−6

x3 8 40 0.35809 4.96 × 10−6 10 42 0.29255 6.59 × 10−6 16 65 0.51380 6.18 × 10−6

x4 17 78 0.59347 7.73 × 10−6 19 79 0.51261 5.79 × 10−6 175 704 4.48920 9.47 × 10−6

x5 32 138 0.98463 7.09 × 10−6 18 75 0.46086 4.05 × 10−6 176 708 4.49410 9.91 × 10−6

x6 17 78 0.57701 9.31 × 10−6 29 120 0.71678 6.05 × 10−6 360 1445 9.10170 9.99 × 10−6

Table 2. Numerical results for MFRM, ACGD and PDY for problem 2 with given initial pointsand dimensions.

MFRM ACGD PDY


1000

x1 3 8 0.007092 5.17 × 10−7 3 8 0.036061 5.17 × 10−7 10 39 0.01053 6.96 × 10−6

x2 3 8 0.012401 6.04 × 10−6 3 8 0.006143 6.04 × 10−6 11 43 0.00937 9.23 × 10−6

x3 4 11 0.003993 4.37 × 10−7 4 11 0.006476 4.37 × 10−7 13 51 0.01111 6.26 × 10−6

x4 5 14 0.010363 1.52 × 10−7 5 14 0.005968 1.52 × 10−7 14 55 0.02154 9.46 × 10−6

x5 5 14 0.007234 1.1 × 10−6 5 14 0.02349 1.1 × 10−6 15 59 0.01850 4.60 × 10−6

x6 6 17 0.006496 1.74 × 10−8 6 17 0.00677 1.74 × 10−8 15 59 0.01938 7.71 × 10−6

5000

x1 3 8 0.011561 1.75 × 10−7 3 8 0.009794 1.75 × 10−7 11 43 0.03528 4.86 × 10−6

x2 3 8 0.010452 3.13 × 10−6 3 8 0.009591 3.13 × 10−6 12 47 0.04032 6.89 × 10−6

x3 4 11 0.01516 1.42 × 10−7 4 11 0.013767 1.42 × 10−7 14 55 0.04889 4.61 × 10−6

x4 5 14 0.019733 3.94 × 10−8 5 14 0.014274 3.94 × 10−8 15 59 0.04826 6.96 × 10−6

x5 5 14 0.018462 4.05 × 10−7 5 14 0.011728 4.05 × 10−7 16 63 0.05969 3.37 × 10−6

x6 6 17 0.028536 2.36 × 10−9 6 17 0.016345 2.36 × 10−9 16 63 0.06253 5.64 × 10−6

10,000

x1 3 8 0.019053 1.21 × 10−7 3 8 0.0135 1.21 × 10−7 11 43 0.06732 6.85 × 10−6

x2 3 8 0.01791 2.79 × 10−6 3 8 0.015807 2.79 × 10−6 12 47 0.12232 9.72 × 10−6

x3 4 11 0.033042 9.73 × 10−8 4 11 0.020752 9.73 × 10−8 14 55 0.08288 6.51 × 10−6

x4 5 14 0.031576 2.56 × 10−8 5 14 0.04483 2.56 × 10−8 15 59 0.08413 9.82 × 10−6

x5 5 14 0.032747 2.93 × 10−7 5 14 0.026975 2.93 × 10−7 16 63 0.09589 4.75 × 10−6

x6 6 17 0.036002 1.24 × 10−9 6 17 0.032445 1.24 × 10−9 16 64 0.11499 8.55 × 10−6

50,000

x1 3 8 0.0737 6.32 × 10−8 7 26 0.16925 2.94 × 10−6 12 47 0.27826 5.23 × 10−6

x2 3 8 0.06964 3.37 × 10−6 9 34 0.18801 2.78 × 10−6 13 51 0.29642 7.11 × 10−6

x3 4 11 0.093027 4.87 × 10−8 7 25 0.15375 9.11 × 10−6 15 59 0.35602 4.82 × 10−6

x4 5 14 0.11219 1.11 × 10−8 7 24 0.15382 9.18 × 10−6 35 141 0.69470 6.69 × 10−6

x5 5 14 0.1173 1.84 × 10−7 9 32 0.18164 6.71 × 10−6 35 141 0.68488 9.12 × 10−6

x6 6 17 0.13794 4.01 × 10−10 6 19 0.11216 5.2 × 10−6 35 141 0.70973 9.91 × 10−6

100,000

x1 3 8 0.13021 5.4 × 10−8 7 26 0.2609 4.14 × 10−6 12 47 0.44541 7.39 × 10−6

x2 3 8 0.13267 4.27 × 10−6 9 34 0.32666 3.93 × 10−6 14 55 0.53299 3.39 × 10−6

x3 4 11 0.17338 4.05 × 10−8 8 29 0.3113 3.33 × 10−6 15 60 0.58603 8.71 × 10−6

x4 5 14 0.20036 8.15 × 10−9 8 28 0.2997 3.34 × 10−6 72 290 2.70630 8.31 × 10−6

x5 5 14 0.25274 1.8 × 10−7 9 32 0.32098 9.46 × 10−6 72 290 2.72220 8.68 × 10−6

x6 6 17 0.24952 2.71 × 10−10 6 19 0.21972 7.01 × 10−6 72 290 2.75850 8.96 × 10−6

316



MFRM ACGD PDY


1000

x1 6 24 0.024062 3.11 × 10−6 6 40 0.02951 4.44 × 10−6 12 48 0.01255 4.45 × 10−6

x2 6 24 0.005345 5.94 × 10−6 6 40 0.0077681 8.75 × 10−6 12 48 0.01311 9.02 × 10−6

x3 6 24 0.006109 9.94 × 10−6 6 44 0.0067049 5.09 × 10−6 13 52 0.01486 8.34 × 10−6

x4 8 33 0.006127 3.1 × 10−6 8 44 0.007142 5.04 × 10−6 14 56 0.01698 8.04 × 10−6

x5 11 46 0.010427 2.71 × 10−6 11 40 0.010411 3.12 × 10−6 14 56 0.01551 9.72 × 10−6

x6 16 68 0.010682 8.38 × 10−6 16 77 0.014759 5.98 × 10−6 14 56 0.01534 9.42 × 10−6

5000

x1 6 24 0.020455 6.96 × 10−6 6 40 0.020368 9.93 × 10−6 12 48 0.03660 9.94 × 10−6

x2 7 28 0.021552 1.33 × 10−6 7 44 0.029622 5.09 × 10−6 13 52 0.03616 6.85 × 10−6

x3 7 28 0.023056 2.22 × 10−6 7 48 0.030044 2.96 × 10−6 14 56 0.04594 6.14 × 10−6

x4 8 33 0.022984 6.92 × 10−6 8 48 0.022777 2.93 × 10−6 15 60 0.04342 6.01 × 10−6

x5 11 46 0.031466 6.06 × 10−6 11 40 0.019226 6.97 × 10−6 15 60 0.04296 7.25 × 10−6

x6 17 72 0.049308 7.67 × 10−6 17 81 0.036095 6.05 × 10−6 32 129 0.10081 8.85 × 10−6

10,000

x1 6 24 0.03064 9.85 × 10−6 6 44 0.03997 3.65 × 10−6 13 52 0.06192 4.77 × 10−6

x2 7 28 0.035806 1.88 × 10−6 7 44 0.037221 7.19 × 10−6 13 52 0.06442 9.68 × 10−6

x3 7 28 0.035795 3.14 × 10−6 7 48 0.053226 4.18 × 10−6 14 56 0.09499 8.69 × 10−6

x4 8 33 0.041017 9.79 × 10−6 8 48 0.057984 4.15 × 10−6 15 60 0.07696 8.5 × 10−6

x5 11 46 0.06448 8.58 × 10−6 11 40 0.047413 9.85 × 10−6 33 133 0.18625 6.45 × 10−6

x6 18 76 0.09651 4.44 × 10−6 18 81 0.085238 8.56 × 10−6 33 133 0.15548 7.51 × 10−6

50,000

x1 7 28 0.14323 2.2 × 10−6 7 44 0.17175 8.17 × 10−6 14 56 0.23642 3.51 × 10−6

x2 7 28 0.13625 4.2 × 10−6 7 48 0.18484 4.18 × 10−6 14 56 0.24813 7.12 × 10−6

x3 7 28 0.13246 7.03 × 10−6 7 48 0.1827 9.36 × 10−6 15 60 0.27049 6.53 × 10−6

x4 9 37 0.18261 4.16 × 10−6 9 48 0.18993 9.27 × 10−6 34 137 0.54545 7.13 × 10−6

x5 12 50 0.21743 5.2 × 10−6 12 44 0.17043 5.73 × 10−6 68 274 1.02330 9.99 × 10−6

x6 18 76 0.34645 9.93 × 10−6 18 85 0.32938 8.66 × 10−6 69 278 1.03810 8.05 × 10−6

100,000

x1 7 28 0.27078 3.11 × 10−6 7 48 0.36144 3 × 10−6 14 56 0.45475 4.96 × 10−6

x2 7 28 0.26974 5.94 × 10−6 7 48 0.37515 5.91 × 10−6 15 60 0.49018 3.39 × 10−6

x3 7 28 0.25475 9.94 × 10−6 7 52 0.39071 3.44 × 10−6 15 60 0.49016 9.24 × 10−6

x4 9 37 0.3089 5.88 × 10−6 9 52 0.35961 3.41 × 10−6 139 559 4.03110 9.01 × 10−6

x5 12 50 0.41839 7.35 × 10−6 12 44 0.33105 8.1 × 10−6 70 282 2.07100 8.54 × 10−6

x6 19 80 0.64773 5.75 × 10−6 19 89 0.61329 5.54 × 10−6 139 559 4.02440 9.38 × 10−6


MFRM ACGD PDY


1000

x1 6 24 0.00855 1.65 × 10−6 10 40 0.014662 3.65 × 10−6 12 48 0.00989 4.60 × 10−6

x2 5 20 0.004234 2.32 × 10−6 10 40 0.0064115 5.79 × 10−6 12 48 0.00966 9.57 × 10−6

x3 10 42 0.007426 6.42 × 10−6 10 40 0.0054818 3.29 × 10−6 13 52 0.00887 8.49 × 10−6

x4 21 90 0.011603 5.84 × 10−6 27 110 0.012854 8.97 × 10−6 12 48 0.01207 5.83 × 10−6

x5 16 71 0.010735 8.48 × 10−6 26 106 0.015603 5.97 × 10−6 29 117 0.05371 9.43 × 10−6

x6 1 15 0.005932 0 36 147 0.025039 9.56 × 10−6 29 117 0.02396 6.65 × 10−6

5000

x1 6 24 0.019995 3.68 × 10−6 10 40 0.018283 8.15 × 10−6 13 52 0.02503 3.49 × 10−6

x2 5 20 0.00934 5.2 × 10−6 11 44 0.016733 3.36 × 10−6 13 52 0.02626 7.24 × 10−6

x3 11 46 0.02156 3.89 × 10−6 10 40 0.017073 7.37 × 10−6 14 56 0.03349 6.29 × 10−6

x4 22 94 0.043325 6.81 × 10−6 29 118 0.047436 7.09 × 10−6 13 52 0.02258 4.25 × 10−6

x5 18 79 0.096692 6.15 × 10−6 27 110 0.058405 7.95 × 10−6 31 125 0.05471 7.59 × 10−6

x6 1 15 0.012199 0 39 159 0.059448 7.33 × 10−6 63 254 0.10064 8.54 × 10−6

10,000

x1 6 24 0.019264 5.2 × 10−6 11 44 0.026877 3 × 10−6 13 52 0.03761 4.93 × 10−6

x2 5 20 0.017891 7.35 × 10−6 11 44 0.03118 4.76 × 10−6 14 56 0.04100 3.37 × 10−6

x3 11 46 0.036079 5.5 × 10−6 11 44 0.034673 2.71 × 10−6 14 56 0.03919 8.90 × 10−6

x4 22 94 0.069778 9.63 × 10−6 30 122 0.069971 5.97 × 10−6 32 129 0.09613 6.02 × 10−6

x5 18 79 0.062821 8.69 × 10−6 28 114 0.066866 6.68 × 10−6 32 129 0.09177 6.44 × 10−6

x6 1 15 0.017237 0 40 163 0.093749 7.26 × 10−6 64 258 0.20791 9.39 × 10−6

50,000

x1 7 28 0.093473 1.16 × 10−6 11 44 0.16749 6.7 × 10−6 14 56 0.17193 3.63 × 10−6

x2 6 24 0.072206 1.64 × 10−6 12 48 0.11391 2.77 × 10−6 14 56 0.15237 7.54 × 10−6

x3 12 50 0.14285 3.33 × 10−6 11 44 0.11036 6.06 × 10−6 15 60 0.16549 6.66 × 10−6

x4 24 102 0.30313 5.86 × 10−6 31 126 0.30903 7.94 × 10−6 67 270 0.76283 7.81 × 10−6

x5 20 87 0.28955 6.31 × 10−6 29 118 0.30266 8.89 × 10−6 67 270 0.76157 8.80 × 10−6

x6 1 15 0.061327 0 42 171 0.41158 7.96 × 10−6 269 1080 2.92510 9.41 × 10−6

100,000

x1 7 28 0.15038 1.65 × 10−6 11 44 0.2434 9.48 × 10−6 14 56 0.30229 5.13 × 10−6

x2 6 24 0.13126 2.32 × 10−6 12 48 0.2614 3.91 × 10−6 15 60 0.31648 3.59 × 10−6

x3 12 50 0.31585 4.71 × 10−6 11 44 0.2161 8.57 × 10−6 32 129 0.72838 9.99 × 10−6

x4 24 102 0.58023 8.29 × 10−6 32 130 0.65289 6.68 × 10−6 135 543 2.86780 9.73 × 10−6

x5 20 87 0.5122 8.92 × 10−6 30 122 0.61637 7.48 × 10−6 272 1092 5.74140 9.91 × 10−6

x6 1 15 0.11696 0 43 175 0.82759 7.88 × 10−6 548 2197 11.44130 9.87 × 10−6

317



MFRM ACGD PDY


1000

x1 26 98 0.023555 3.51 × 10−6 39 154 0.022285 9.7 × 10−6 16 63 0.07575 6.03 × 10−6

x2 40 154 0.024539 5.9 × 10−6 22 85 0.015671 5.03 × 10−6 16 63 0.01470 5.42 × 10−6

x3 37 144 0.021659 7.11 × 10−6 43 173 0.029569 7.96 × 10−6 33 132 0.02208 6.75 × 10−6

x4 49 206 0.030696 9.52 × 10−6 30 122 0.014942 6.05 × 10−6 30 121 0.01835 8.39 × 10−6

x5 46 194 0.11589 7.06 × 10−6 29 118 0.040406 6.5 × 10−6 32 129 0.02700 8.47 × 10−6

x6 43 182 0.027471 8.7 × 10−6 40 163 0.0311 9.83 × 10−6 30 121 0.01712 6.95 × 10−6

5000

x1 38 147 0.073315 4.96 × 10−6 30 117 0.060877 9.56 × 10−6 17 67 0.04394 5.64 × 10−6

x2 20 77 0.056225 4.98 × 10−6 16 60 0.027911 5.91 × 10−6 17 67 0.04635 5.07 × 10−6

x3 41 157 0.082151 8.92 × 10−6 78 315 0.12774 9.7 × 10−6 35 140 0.08311 9.74 × 10−6

x4 48 202 0.10166 9.19 × 10−6 31 126 0.067911 8.39 × 10−6 33 133 0.08075 6.02 × 10−6

x6 147 562 3.308158 8.44 × 10−7 31 126 0.067856 7.81 × 10−6 35 141 0.10091 7.51 × 10−6

x7 45 190 0.090276 7.14 × 10−6 44 179 0.09371 7.37 × 10−6 32 129 0.08054 8.55 × 10−6

10,000

x1 37 143 0.12665 9.28 × 10−6 77 308 0.28678 9.85 × 10−6 17 67 0.06816 8.81 × 10−6

x2 22 84 0.077288 9.78 × 10−6 16 60 0.071657 7.52 × 10−6 17 67 0.08833 7.80 × 10−6

x3 39 149 0.1297 6.74 × 10−6 105 424 0.34212 9.08 × 10−6 37 148 0.14732 6.36 × 10−6

x4 60 250 0.2175 7.56 × 10−6 32 130 0.11937 7.17 × 10−6 37 149 0.14293 8.25 × 10−6

x5 44 186 0.1727 7.68 × 10−6 32 130 0.11921 8.26 × 10−6 36 145 0.14719 8.23 × 10−6

x6 46 194 0.1728 8.62 × 10−6 45 183 0.15634 9.01 × 10−6 74 298 0.26456 7.79 × 10−6

50,000

x1 44 170 0.62202 1 × 10−5 90 539 31.75299 2.56 × 10−7 42 169 0.58113 7.78 × 10−6

x2 69 280 0.9662 6.87 × 10−6 31 122 0.33817 7.09 × 10−6 42 169 0.58456 7.13 × 10−6

x3 119 464 25.87657 9.34 × 10−7 260 1047 2.8824 9.67 × 10−6 41 165 0.58717 8.87 × 10−6

x4 50 210 0.71599 8.38 × 10−6 33 134 0.39039 9.98 × 10−6 40 161 0.56431 7.17 × 10−6

x5 46 194 0.65538 8.47 × 10−6 35 142 0.40807 7.19 × 10−6 82 330 1.08920 8.44 × 10−6

x6 50 210 0.69117 8.12 × 10−6 49 199 0.57702 8.97 × 10−6 80 322 1.06670 7.82 × 10−6

100,000

x1 31 121 0.84183 4.48 × 10−6 88 530 61.97806 5.53 × 10−7 43 173 1.09620 8.47 × 10−6

x2 135 518 59.19294 8.37 × 10−7 110 442 2.2661 9.55 × 10−6 43 173 1.10040 7.77 × 10−6

x3 46 178 1.1322 6.99 × 10−6 345 1388 7.1938 9.76 × 10−6 42 169 1.08330 9.66 × 10−6

x4 50 210 1.3737 8.85 × 10−6 34 138 0.74362 8.65 × 10−6 85 342 2.11880 9.22 × 10−6

x5 47 198 1.3879 8.31 × 10−6 36 146 0.79012 8.09 × 10−6 84 338 2.10640 9.78 × 10−6

x6 52 218 1.4318 7.37 × 10−6 51 207 1.1601 8.42 × 10−6 167 671 4.06200 9.90 × 10−6

Table 6. Numerical Results for MFRM, ACGD and PDY for problem 6 with given initial pointsand dimensions.

MFRM ACGD PDY


1000

x1 11 44 0.011156 8.32 × 10−6 12 48 0.02786 7.88 × 10−6 15 60 0.01671 4.35 × 10−6

x2 11 44 0.016092 7.32 × 10−6 12 48 0.01042 7.58 × 10−6 15 60 0.01346 4.18 × 10−6

x3 11 44 0.010446 8.83 × 10−6 12 48 0.0092 6.68 × 10−6 15 60 0.01630 3.68 × 10−6

x4 10 40 0.011233 7.38 × 10−6 12 48 0.013617 4.57 × 10−6 14 56 0.01339 7.48 × 10−6

x5 9 36 0.011325 8.29 × 10−6 12 48 0.011492 3.67 × 10−6 14 56 0.01267 6.01 × 10−6

x6 7 28 0.009452 8.25 × 10−6 11 44 0.016351 8.32 × 10−6 14 56 0.01685 3.54 × 10−6

5000

x1 8 32 0.026924 1.87 × 10−6 13 52 0.036025 4.59 × 10−6 15 60 0.05038 9.73 × 10−6

x2 8 32 0.043488 1.8 × 10−6 13 52 0.040897 4.42 × 10−6 15 60 0.04775 9.36 × 10−6

x3 8 32 0.02709 1.59 × 10−6 13 52 0.039937 3.89 × 10−6 15 60 0.04923 8.25 × 10−6

x4 8 32 0.026351 1.1 × 10−6 13 52 0.033013 2.66 × 10−6 15 60 0.05793 5.64 × 10−6

x5 7 28 0.023442 8.62 × 10−6 12 48 0.030462 8.22 × 10−6 15 60 0.04597 4.53 × 10−6

x6 7 28 0.022952 5.08 × 10−6 12 48 0.028786 4.85 × 10−6 14 56 0.05070 7.93 × 10−6

10,000

x1 8 32 0.061374 2.62 × 10−6 13 52 0.092372 6.5 × 10−6 68 274 0.40724 9.06 × 10−6

x2 8 32 0.06285 2.52 × 10−6 13 52 0.059778 6.25 × 10−6 68 274 0.41818 8.72 × 10−6

x3 8 32 0.059913 2.22 × 10−6 13 52 0.077326 5.5 × 10−6 34 137 0.21905 6.22 × 10−6

x4 8 32 0.057003 1.52 × 10−6 13 52 0.087745 3.77 × 10−6 15 60 0.10076 7.98 × 10−6

x5 8 32 0.070377 1.22 × 10−6 13 52 0.077217 3.02 × 10−6 15 60 0.12680 6.40 × 10−6

x6 7 28 0.052718 7.18 × 10−6 12 48 0.067375 6.85 × 10−6 15 60 0.11984 3.78 × 10−6

50,000

x1 8 32 0.21258 5.85 × 10−6 14 56 0.32965 3.78 × 10−6 143 575 3.09120 9.42 × 10−6

x2 8 32 0.21203 5.63 × 10−6 14 56 0.31297 3.63 × 10−6 143 575 3.06200 9.06 × 10−6

x3 8 32 0.20885 4.96 × 10−6 14 56 0.30089 3.2 × 10−6 142 571 3.04950 9.04 × 10−6

x4 8 32 0.20483 3.4 × 10−6 13 52 0.26855 8.42 × 10−6 69 278 1.53920 9.14 × 10−6

x5 8 32 0.21467 2.72 × 10−6 13 52 0.26304 6.76 × 10−6 68 274 1.49490 9.43 × 10−6

x6 8 32 0.20933 1.61 × 10−6 13 52 0.26143 3.99 × 10−6 15 60 0.38177 8.44 × 10−6

100,000

x1 8 32 0.41701 8.28 × 10−6 14 56 0.58853 5.34 × 10−6 292 1172 13.59530 9.53 × 10−6

x2 8 32 0.41511 7.96 × 10−6 14 56 0.58897 5.14 × 10−6 290 1164 13.30930 9.75 × 10−6

x3 8 32 0.44061 7.01 × 10−6 14 56 0.57318 4.53 × 10−6 144 579 6.68150 9.96 × 10−6

x4 8 32 0.43805 4.8 × 10−6 14 56 0.58712 3.1 × 10−6 141 567 6.50800 9.92 × 10−6

x5 8 32 0.41147 3.85 × 10−6 13 52 0.56384 9.56 × 10−6 70 282 3.30510 8.07 × 10−6

x6 8 32 0.43925 2.27 × 10−6 13 52 0.53343 5.64 × 10−6 34 137 1.64510 6.37 × 10−6

318


Table 7. Numerical Results for MFRM, ACGD and PDY for problem 7 with given initial pointsand dimensions.

MFRM ACGD PDY


1000

x1 4 21 0.011834 3.24 × 10−7 10 42 0.008528 2.46 × 10−6 14 57 0.00953 5.28 × 10−6

x2 4 21 0.006228 1.43 × 10−7 9 38 0.008289 3.91 × 10−6 13 53 0.00896 9.05 × 10−6

x3 3 17 0.004096 5.81 × 10−8 8 34 0.006702 7.43 × 10−6 3 12 0.00426 8.47 × 10−6

x4 7 34 0.00585 3.89 × 10−6 11 46 0.009579 5.94 × 10−6 15 61 0.01169 6.73 × 10−6

x5 7 34 0.006133 6.36 × 10−6 11 46 0.015328 8.97 × 10−6 31 126 0.03646 9.03 × 10−6

x6 8 37 0.006106 1.9 × 10−6 12 49 0.01426 2.87 × 10−6 15 60 0.01082 3.99 × 10−6

5000

x1 4 21 0.015836 7.25 × 10−7 10 42 0.023953 5.49 × 10−6 15 61 0.03215 4.25 × 10−6

x2 4 21 0.014521 3.2 × 10−7 9 38 0.021065 8.74 × 10−6 14 57 0.02942 7.40 × 10−6

x3 3 17 0.014517 1.3 × 10−7 9 38 0.025437 4.01 × 10−6 4 16 0.01107 1.01 × 10−7

x4 7 34 0.028388 8.71 × 10−6 12 50 0.028607 3.21 × 10−6 16 65 0.04331 5.43 × 10−6

x5 8 38 0.02787 1.49 × 10−6 12 50 0.037806 4.84 × 10−6 33 134 0.09379 7.78 × 10−6

x6 8 37 0.027898 4.26 × 10−6 12 49 0.029226 6.43 × 10−6 15 60 0.04077 8.92 × 10−6

10,000

x1 4 21 0.028528 1.02 × 10−6 10 42 0.045585 7.77 × 10−6 15 61 0.06484 6.01 × 10−6

x2 4 21 0.033782 4.52 × 10−7 10 42 0.041715 2.98 × 10−6 15 61 0.07734 3.77 × 10−6

x3 3 17 0.029265 1.84 × 10−7 9 38 0.036422 5.67 × 10−6 4 16 0.02707 1.42 × 10−7

x4 8 38 0.043301 1.29 × 10−6 12 50 0.063527 4.53 × 10−6 16 65 0.07941 7.69 × 10−6

x5 8 38 0.043741 2.1 × 10−6 12 50 0.049604 6.85 × 10−6 34 138 0.14942 6.83 × 10−6

x6 8 37 0.053666 6.02 × 10−6 12 49 0.050153 9.09 × 10−6 34 138 0.15224 8.81 × 10−6

50,000

x1 4 21 0.10816 2.29 × 10−6 11 46 0.20624 4.19 × 10−6 16 65 0.25995 4.89 × 10−6

x2 4 21 0.11969 1.01 × 10−6 10 42 0.16364 6.67 × 10−6 15 61 0.24674 8.42 × 10−6

x3 3 17 0.068644 4.11 × 10−7 10 42 0.1539 3.06 × 10−6 4 16 0.09405 3.18 × 10−7

x4 8 38 0.16067 2.88 × 10−6 13 54 0.20728 2.45 × 10−6 36 146 0.55207 6.39 × 10−6

x5 8 38 0.14484 4.7 × 10−6 13 54 0.19421 3.69 × 10−6 35 142 0.54679 9.05 × 10−6

x6 9 41 0.161 1.41 × 10−6 13 53 0.19386 4.9 × 10−6 36 146 0.55764 7.59 × 10−6

100,000

x1 4 21 0.21825 3.24 × 10−6 11 46 0.32512 5.93 × 10−6 17 69 0.52595 5.68 × 10−6

x2 4 21 0.16435 1.43 × 10−6 10 42 0.30949 9.43 × 10−6 16 65 0.52102 4.34 × 10−6

x3 3 17 0.13072 5.81 × 10−7 10 42 0.31031 4.32 × 10−6 4 16 0.14864 4.50 × 10−7

x4 8 38 0.29012 4.07 × 10−6 13 54 0.38833 3.46 × 10−6 36 146 1.05360 9.04 × 10−6

x5 8 38 0.32821 6.65 × 10−6 13 54 0.3522 5.22 × 10−6 74 299 2.10730 8.55 × 10−6

x6 9 41 0.43649 1.99 × 10−6 13 53 0.3561 6.94 × 10−6 37 150 1.08240 6.66 × 10−6


MFRM ACGD PDY


1000

x1 8 27 0.1502 1.52 × 10−6 8 26 0.049826 6.09 × 10−6 69 279 0.05538 8.95 × 10−6

x2 8 27 0.042248 1.52 × 10−6 8 26 0.017594 6.09 × 10−6 270 1085 0.18798 9.72 × 10−6

x3 26 114 0.03877 7.85 × 10−6 8 26 0.010888 6.09 × 10−6 24 52 0.02439 6.57 × 10−6

x4 26 114 0.017542 7.85 × 10−6 8 26 0.007873 6.09 × 10−6 27 58 0.01520 7.59 × 10−6

x5 26 114 0.067692 7.85 × 10−6 8 26 0.060733 6.09 × 10−6 28 61 0.04330 9.21 × 10−6

x6 26 114 0.045173 7.85 × 10−6 8 26 0.006889 6.09 × 10−6 40 85 0.02116 8.45 × 10−6

5000

x1 6 28 0.023925 8.77 × 10−6 4 13 0.011005 5.76 × 10−6 658 2639 1.13030 9.98 × 10−6

x2 15 70 0.043512 7.94 × 10−6 4 13 0.009131 5.76 × 10−6 27 58 0.05101 7.59 × 10−6

x3 15 70 0.046458 7.94 × 10−6 4 13 0.011311 5.76 × 10−6 49 104 0.08035 8.11 × 10−6

x4 15 70 0.044788 7.94 × 10−6 4 13 0.010475 5.75 × 10−6 40 85 0.07979 8.45 × 10−6

x5 15 70 0.044639 7.94 × 10−6 4 13 0.011034 5.77 × 10−6 18 40 0.09128 9.14 × 10−6

x6 15 70 0.043974 7.94 × 10−6 4 13 0.00785 5.76 × 10−6 17 38 0.18528 8.98 × 10−6

10,000

x1 11 54 0.06595 6.15 × 10−6 5 20 0.024232 2.19 × 10−6 49 104 0.20443 7.62 × 10−6

x2 11 54 0.068125 6.15 × 10−6 5 20 0.023511 2.19 × 10−6 40 85 0.15801 8.45 × 10−6

x3 11 54 0.065486 6.15 × 10−6 5 20 0.023004 2.19 × 10−6 19 42 0.37880 7.66 × 10−6

x4 11 54 0.064515 6.15 × 10−6 5 20 0.030435 2.19 × 10−6 90 187 1.25802 9.7 × 10−6

x5 11 54 0.056261 6.15 × 10−6 5 20 0.021963 2.19 × 10−6 988 1988 12.68259 9.93 × 10−6

x6 11 54 0.067785 6.15 × 10−6 5 20 0.021889 2.21 × 10−6 27 58 0.32859 7.59 × 10−6

50,000

x1 7 38 0.17856 4.5 × 10−6 5 23 0.087544 2.45 × 10−6 19 42 0.52291 6.42 × 10−6

x2 7 38 0.17862 4.5 × 10−6 5 23 0.093227 2.45 × 10−6 148 304 3.93063 9.92 × 10−6

x3 7 38 0.17746 4.5 × 10−6 5 23 0.087484 2.45 × 10−6 937 1886 22.97097 9.87 × 10−6

x4 7 38 0.17392 4.5 × 10−6 5 23 0.086329 2.4 × 10−6 27 58 0.68467 7.59 × 10−6

x5 7 38 0.18035 4.5 × 10−6 5 23 0.08954 2.4 × 10−6 346 702 8.45043 9.79 × 10−6

x6 7 38 0.17504 4.5 × 10−6 5 23 0.093203 2.5 × 10−6 40 85 0.99230 8.45 × 10−6

100,000

x1 28 122 0.91448 8.61 × 10−6 4 20 0.14743 2.71 × 10−6 - - - -x2 28 122 0.93662 8.61 × 10−6 4 20 0.14823 2.7 × 10−6 - - - -x3 28 122 0.90604 8.61 × 10−6 4 20 0.1497 2.79 × 10−6 - - - -x4 28 122 0.92351 8.61 × 10−6 4 20 0.14844 2.37 × 10−6 - - - -x5 28 122 0.91896 8.61 × 10−6 4 20 0.12346 1.66 × 10−6 - - - -x6 28 122 0.91294 8.61 × 10−6 4 20 0.12522 2.11 × 10−6 - - - -

Figure 1 reveals that MFRM performed better in terms of number of Iterations, as it solves andwins over 70 percent of the problems with less number of Iterations, while ACGD and PDY solve andwin over 40 and almost 10 percent respectively. The story is a little bit different in Figure 2 as ACGDmethod was very competitive. However, MFRM method performed a little bit better by solving andwinning over 50 percent of the problems with less CPU time as against ACGD method which solves

319


and wins less than 50 percent of the problems considered. The PDY method had the least performancewith just 10 percent success. The interpretation of Figure 3 was similar to that of Figure 1. Finally,in Table 11 we report numerical results for MFRM, ACGD and PDY for problem 2 with given initialpoints and dimensions with double float (10−16) accuracy.

Figure 1. Performance profiles for the number of iterations.

Figure 2. Performance profiles for the CPU time (in seconds).

320


Figure 3. Performance profiles for the number of function evaluations.

4.1. Experiments on Solving Sparse Signal Problems

There were many problems in signal processing and statistical inference involving finding sparsesolutions to ill-conditioned linear systems of equations. Among popular approaches was minimizingan objective function which contains quadratic (�2) error term and a sparse �1−regularization term, i.e.,

minx

12‖y− Bx‖2

2 + η‖x‖1, (25)

where x ∈ Rn, y ∈ Rk is an observation, B ∈ Rk×n (k << n) is a linear operator, η is a non-negativeparameter, ‖x‖2 denotes the Euclidean norm of x and ‖x‖1 = ∑n

i=1 |xi| is the �1−norm of x. It is easyto see that problem (25) is a convex unconstrained minimization problem. Due to the fact that if theoriginal signal is sparse or approximately sparse in some orthogonal basis, problem (25) frequentlyappears in compressive sensing, and hence an exact restoration can be produced by solving (25).

Iterative methods for solving (25) have been presented in many papers (see [42–45]). The mostpopular method among these methods is the gradient-based method and the earliest gradient projectionmethod for sparse reconstruction (GPRS) was proposed by Figueiredo et al. [44]. The first step ofthe GPRS method is to express (25) as a quadratic problem using the following process. Considera point x ∈ Rn such that x = u − v, where u, v ≥ 0. u and v are chosen in such a way that x issplitted into its positive and negative parts as follows ui = (xi)+, vi = (−xi)+ for all i = 1, 2, ..., n,and (.)+ = max{0, .}. By definition of �1-norm, we have ‖x‖1 = eT

n u + eTn v, where en = (1, 1, ..., 1)T ∈

Rn. Now (25) can be written as

minu,v

12‖y− B(u− v)‖2

2 + ηeTn u + ηeT

n v, u ≥ 0, v ≥ 0, (26)

which is a bound-constrained quadratic program. However, from [44], Equation (26) can be written instandard form as

minz

12

zT Dz + cTz, such that z ≥ 0, (27)

321


where z =

(uv

), c = ωe2n +

(−bb

), b = BTy, D =

(BT B −BT B−BT B BT B

). Clearly, D is a positive

semi-definite matrix, which implies that Equation (27) is a convex quadratic problem.Xiao et al. [20] translated (27) into a linear variable inequality problem which is equivalent to a

linear complementarity problem. Moreover, z is a solution of the linear complementarity problem ifand only if it is a solution of the following nonlinear equation:

F(z) = min{z, Dz + c} = 0. (28)

The function F is a vector-valued function and the “min” was interpreted as component wiseminimum. Furthermore, F was proved to be continuous and monotone in [46]. Therefore problem (25)can be translated into problem (1) and thus MFRM method can be applied to solve it.

In this experiment, we consider a simple compressive sensing possible situation, where our goalis to reconstruct a sparse signal of length n from k observations. The quality of recovery is assessed bymean of squared error (MSE) to the original signal x,

MSE =1n‖x− x∗‖2,

where x∗ is the recovered signal. The signal size is chosen as n = 211, k = 29 and the original signalcontains 26 randomly nonzero elements. In addition, the measurement y is distributed with noise,that is, y = Bx + �, where B is a randomly generated Gaussian matrix and � is the Gaussian noisedistributed normally with mean 0 and variance 10−4.

To demonstrate the performance of the MFRM method in signal recovery problems, wecompare it with the conjugate gradient descent CGD [20] and projected conjugate gradientPCG [23] methods. The parameters in PCG and CGD methods are chosen as γ = 10, σ = 10−4,ρ = 0.5. However, we chose γ = 1, σ = 10−4, ρ = 0.9 and μ = 0.01 in MFRM method.For fairness in comparison, each code was run from the same initial point, same continuationtechnique on the parameter η, and observed only the behavior of the convergence of each methodto have a similar accurate solution. The experiment was initialized with x0 = BTy and terminates when

‖ f (xk)− f (xk−1)‖‖ f (xk−1)‖ < 10−5,

where f (xk) =12‖y− Bxk‖2

2 + η‖xk‖1.In Figures 4 and 5, MFRM, CGD and PCG methods recovered the disturbed signal almost exactly.

The experiment was repeated for 20 different noise samples (see Table 9). It can be observed thatthe MFRM is more efficient in terms of the number of Iterations and CPU time than CGD and PCGmethods in most cases. Furthermore, MFRM was able to achieve the least MSE in nine (9) out of thetwenty (20) experiments. To reveal visually the performance of both methods, two figures were plottedto demonstrate their convergence behavior based on MSE, objective function values, the numberof Iterations and CPU time (see Figures 6 and 7). It can also be observed that MFRM requires lesscomputing time to achieve similar quality resolution. This can be seen graphically in Figures 6 and 7which illustrate that the objective function values obtained by MFRM decrease faster throughout theentire Iteration process.

322


Figure 4. (top) to (bottom) The original image, the measurement, and the recovered signals by projectedconjugate gradient PCG and modified descent Fletcher–Reeves CG method (MFRM) methods.

Figure 5. (top) to (bottom) The original image, the measurement, and the recovered signals byconjugate gradient descent (CGD) and MFRM methods.

323


Figure 6. Comparison result of PCG and MFRM. The x-axis represent the number of Iterations((top left) and (bottom left)) and CPU time in seconds ((top right) and (bottom right)). The y-axisrepresent the MSE ((top left) and (top right)) and the objective function values ((bottom left) and(bottom right)).

Table 9. Twenty experiment results of �1−norm regularization problem for CGD, PCG andMFRM methods.

S/N Iter Time MSE

CGD PCG MFRM CGD PCG MFRM CGD PCG MFRM

1 248 138 98 2.28 1.28 1.33 6.16 × 10−5 6.32 × 10−5 1.97 × 10−5

2 234 138 117 3.37 1.26 1.19 4.08 × 10−5 3.36 × 10−5 5.40 × 10−5

3 224 152 104 1.90 1.29 0.97 2.78 × 10−5 1.78 × 10−5 1.02 × 10−5

4 230 143 117 3.21 2.48 1.17 4.08 × 10−5 3.36 × 10−5 5.40 × 10−5

5 152 119 114 1.65 1.03 1.15 1.23 × 10−5 2.07 × 10−5 5.49 × 10−5

6 223 127 110 1.89 2.56 1.83 3.33 × 10−5 6.08 × 10−5 6.50 × 10−6

7 156 120 125 1.37 1.01 1.20 4.25 × 10−5 3.26 × 10−5 1.46 × 10−5

8 213 89 10 1.90 0.78 1.12 1.86 × 10−5 3.77 × 10−4 1.31 × 10−5

9 227 152 118 2.14 1.53 1.45 2.75 × 10−5 1.54 × 10−5 8.11 × 10−6

10 201 142 101 2.22 1.64 1.01 6.75 × 10−5 1.86 × 10−5 1.17 × 10−5

11 200 151 90 1.70 1.42 0.90 2.36 × 10−5 1.29 × 10−5 3.81 × 10−5

12 202 153 91 1.75 1.34 0.84 6.94 × 10−5 2.99 × 10−5 9.21 × 10−5

13 208 128 125 1.89 1.12 1.26 1.71 × 10−5 1.42 × 10−5 9.20 × 10−6

14 161 145 122 1.47 1.28 1.26 1.15 × 10−5 8.75 × 10−6 4.36 × 10−6

15 227 160 100 1.97 1.42 1.00 3.41 × 10−5 2.40 × 10−5 1.54 × 10−5

16 269 172 88 2.51 1.67 0.98 3.90 × 10−5 6.59 × 10−5 2.08 × 10−4

17 210 129 105 1.84 1.19 1.11 2.11 × 10−5 1.89 × 10−5 6.22 × 10−5

18 225 132 96 1.93 1.15 1.00 3.87 × 10−5 7.78 × 10−5 9.49 × 10−5

19 152 120 92 1.37 1.09 0.87 2.12 × 10−5 1.32 × 10−5 4.03 × 10−5

20 151 128 113 1.31 1.15 1.06 4.48 × 10−5 1.85 × 10−5 1.71 × 10−5

324


Figure 7. Comparison result of PCG and MFRM. The x-axis represent the number of Iterations((top left) and (bottom left)) and CPU time in seconds ((top right) and (bottom right)). The y-axisrepresent the MSE ((top left) and (top right)) and the objective function values ((bottom left) and(bottom right)).

4.2. Experiments on Blurred Image Restoration

In this subsection, we test the performance of MFRM in restoring a blurred image. We use thefollowing well-known gray test images; (P1) Cameraman, (P2) Lena, (P3) House and (P4) Peppers forthe experiments. We use 4 different Gaussian blur kernels with a standard deviation υ to compare therobustness of MFRM method with CGD method proposed in [20].

To assess the performance of each algorithm tested with respect to the metrics that indicate betterquality of restoration, in Table 10 we reported the objective function (ObjFun) at the approximatesolution, the MSE, the signal-to-noise-ratio (SNR) which is defined as

SNR = 20× log10( ‖x‖‖x− x‖

),

and the structural similarity (SSIM) index that measure the similarity between the original image andthe restored image [47] for each of the 16 experiments. The MATLAB implementation of the SSIMindex can be obtained at http://www.cns.nyu.edu/~lcv/ssim/.

The original, blurred and restored images by each of the algorithm are given in Figures 8–11.The figures demonstrate that both the two algorithms can restore the blurred images. In contrast tothe CGD, the quality of the restored image by MFRM is superior in most cases. Table 11 reportednumerical results for MFRM, ACGD and PDY for problem 2.

325


Table 10. Efficiency comparison based on the value of the objective function (ObjFun)mean-square-error (MSE), SNR and the SSIM index under different Pi(υ).

Image ObjFun MSE SNR SSIM

MFRM CGD MFRM CGD MFRM CGD MFRM CGD

P1(1 × 10−4) 1.43 × 106 1.47 × 106 133.90 177.57 21.28 20.05 0.86 0.83P1(1 × 10−1) 1.43 × 106 1.48 × 106 130.60 177.69 21.39 20.5 0.86 0.83

P1(0.25) 1.47 × 106 1.48 × 106 145.27 177.72 20.93 20.05 0.85 0.83P1(6.25) 1.58 × 106 1.65 × 106 146.06 183.96 20.9 19.9 0.75 0.79

P2(1 × 10−4) 1.61 × 106 1.65 × 106 36.88 57.55 27.59 25.65 0.88 0.86P2(1 × 10−1) 1.61 × 106 1.65 × 106 36.85 57.61 27.59 25.65 0.88 0.86

P2(0.25) 1.62 × 106 1.66 × 106 37.78 57.68 27.48 25.64 0.88 0.86P2(6.25) 1.77 × 106 1.82 × 106 56.65 58.96 25.72 25.55 0.76 0.83

P3(1 × 10−4) 5.74 × 106 5.89 × 106 41.63 44.48 26.26 25.97 0.9 0.88P3(1 × 10−1) 5.75 × 106 5.90 × 106 42.42 44.54 26.17 25.96 0.89 0.88

P3(0.25) 5.76 × 106 5.91 × 106 43.33 44.65 26.08 25.95 0.88 0.88P3(6.25) 6.35 × 106 6.60 × 106 106.79 48.47 22.16 25.6 0.63 0.85

P4(1 × 10−4) 1.40 × 106 1.48 × 106 88.81 122.44 22.9 21.5 0.87 0.84P4(1 × 10−1) 1.41 × 106 1.48 × 106 89.22 122.56 22.88 21.5 0.87 0.84

P4(0.25) 1.41 × 106 1.49 × 106 89.86 122.56 22.85 21.5 0.87 0.84P4(6.25) 1.56 × 106 1.69 × 106 116.79 138.97 21.71 20.95 0.76 0.82

Table 11. Numerical results for modified Fletcher-Reeves method MFRM, accelerated conjugategradient descent (ACGD) and projected Dai-Yuan (PDY) methods for problem 2 with given initialpoints and dimensions with double float (10−16) accuracy.

MFRM ACGD PDY


1000

x1 8 27 0.14061 9.47 × 10−19 12 53 0.030479 3.32 × 10−18 30 119 0.04027 4.76 × 10−19

x2 8 36 0.010782 1.49 × 10−18 7 20 0.013503 1.08 × 10−18 36 153 0.034454 3.51 × 10−18

x3 7 20 0.008263 1.21 × 10−18 13 56 0.021302 3.26 × 10−18 38 161 0.038168 3.51 × 10−18

x4 8 23 0.015654 1.80 × 10−19 12 51 0.02056 3.31 × 10−18 39 165 0.057793 3.51 × 10−18

x5 11 38 0.018461 1.59 × 10−18 14 59 0.088858 3.34 × 10−18 41 173 0.069756 3.51 × 10−18

x6 10 34 0.016788 1.07 × 10−18 10 32 0.012069 5.83 × 10−19 40 169 0.03311 3.50 × 10−18

5000

x1 9 33 0.028658 7.22 × 10−19 12 54 0.041685 1.52 × 10−18 35 149 0.10692 1.57 × 10−18

x2 7 23 0.024046 2.18 × 10−19 9 41 0.049194 1.55 × 10−18 37 157 0.12219 1.57 × 10−18

x3 6 17 0.03436 3.89 × 10−19 14 61 0.094129 1.47 × 10−18 33 131 0.10635 1.06 × 10−19

x4 8 26 0.03133 7.17 × 10−19 14 60 0.065147 1.47 × 10−18 39 165 0.18361 1.57 × 10−18

x5 9 31 0.036727 5.84 × 10−19 10 43 0.1165 1.47 × 10−18 36 144 0.2178 7.43 × 10−20

x6 10 34 0.030168 6.41 × 10−19 12 51 0.038218 1.51 × 10−18 38 161 0.13144 1.57 × 10−18

10,000

x1 8 28 0.064617 1.89 × 10−19 11 50 0.068567 1.03 × 10−18 35 149 0.2253 1.11 × 10−18

x2 6 19 0.044204 1.90 × 10−19 14 62 0.15949 1.09 × 10−18 32 128 0.34325 8.21 × 10−20

x3 6 17 0.045192 1.45 × 10−19 18 78 0.10766 1.04 × 10−18 39 165 0.23899 1.11 × 10−18

x4 10 35 0.055408 4.99 × 10−19 12 52 0.061589 1.06 × 10−18 39 165 0.23162 1.11 × 10−18

x5 7 20 0.038439 2.06 × 10−19 14 60 0.087394 1.05 × 10−18 40 169 0.28998 1.11 × 10−18

x6 9 29 0.065318 5.27 × 10−19 16 68 0.09917 1.03 × 10−18 40 170 0.22564 1.11 × 10−18

50,000

x1 7 26 0.21017 1.93 × 10−19 23 100 0.51879 4.79 × 10−19 34 145 0.92896 4.96 × 10−19

x2 6 21 0.24752 2.09 × 10−19 25 108 0.64677 4.90 × 10−19 36 153 0.9954 4.96 × 10−19

x3 6 17 0.11243 6.27 × 10−20 23 99 0.50402 4.93 × 10−19 38 161 0.96768 4.96 × 10−19

x4 7 20 0.13442 1.02 × 10−19 24 102 0.63664 4.75 × 10−19 79 326 1.7542 4.96 × 10−19

x5 9 30 0.20288 7.25 × 10−20 25 106 0.51116 4.78 × 10−19 78 322 1.7246 4.96 × 10−19

x6 12 52 0.36526 2.28 × 10−19 23 97 0.56342 4.76 × 10−19 80 330 1.6812 4.96 × 10−19

100,000

x1 7 27 0.36065 6.53 × 10−20 23 100 0.88236 3.26 × 10−19 30 119 1.2102 9.26 × 10−21

x2 5 14 0.20041 3.91 × 10−20 25 108 0.90777 3.27 × 10−19 35 149 1.5699 3.51 × 10−19

x3 7 24 0.34075 1.47 × 10−19 25 107 0.95898 3.26 × 10−19 40 170 1.7126 3.51 × 10−19

x4 8 31 0.40444 2.09 × 10−20 24 102 0.83332 3.38 × 10−19 151 614 5.8306 3.51 × 10−19

x5 8 26 0.52598 5.03 × 10−20 25 106 1.0223 3.47 × 10−19 151 614 5.6777 3.50 × 10−19

x6 7 20 0.33434 1.45 × 10−19 23 97 0.87438 3.33 × 10−19 153 622 5.7906 3.51 × 10−19

326


Figure 8. The original image (top left), the blurred image (top right), the restored image by CGD(bottom left) with time = 3.70, signal-to-noise-ratio (SNR) = 20.05 and structural similarity (SSIM)= 0.83, and by MFRM (bottom right) with time = 1.97, SNR = 21.28 and SSIM = 0.86.

Figure 9. The original image (top left), the blurred image (top right), the restored image by CGD(bottom left) with Time = 1.95, SNR = 25.65 and SSIM = 0.86, and by MFRM (bottom right) withTime = 3.59, SNR = 27.59 and SSIM = 0.88.

327


Figure 10. The original image (top left), the blurred image (top right), the restored image by CGD(bottom left) with time = 5.38, SNR = 25.97 and SSIM = 0.88, and by MFRM (bottom right) with time= 38.77, SNR = 26.26 and SSIM = 0.90.

Figure 11. The original image (top left), the blurred image (top right), the restored image by CGD(bottom left) with Time = 2.48, SNR = 21.50 and SSIM = 0.84, and by MFRM (bottom right) withTime = 4.93, SNR = 22.90 and SSIM = 0.87.

328


5. Conclusions

In this paper, a modified conjugate gradient method for solving monotone nonlinear equationswith convex constraints was presented which is similar to that in [3]. The proposed method is suitablefor non-smooth equations. Under some suitable assumptions, the global convergence of the proposedmethod was demonstrated. Numerical results were presented to show the effectiveness of the MFRMmethod compared to the ACGD and PDY methods for the given constrained monotone equationproblems. Finally, the MFRM was also shown to be effective in decoding sparse signals and restorationof blurred images.

Author Contributions: conceptualization, A.B.A.; methodology, A.B.A.; software, H.M.; validation, P.K., A.M.A.and K.S.; formal analysis, P.K. and K.S.; investigation, P.K. and H.M.; resources, P.K. and K.S.; data curation, H.M.and A.M.A.; writing–original draft preparation, A.B.A.; writing–review and editing, H.M.; visualization, A.M.A.and K.S.; supervision, P.K.; project administration, P.K. and K.S.; funding acquisition, P.K. and K.S.

Funding: Petchra Pra Jom Klao Doctoral Scholarship for Ph.D. program of King Mongkut’s University ofTechnology Thonburi (KMUTT). This project was partially supported by the Thailand Research Fund (TRF) andthe King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (GrantNo. RSA6080047). Moreover, Kanokwan Sitthithakerngkiet was supported by Faculty of Applied Science, KingMongkuts University of Technology North Bangkok. Contract no. 6242104.

Acknowledgments: We thank Associate Professor Jin Kiu Liu for providing us with the access of the CGD-CSMATLAB codes. The authors acknowledge the financial support provided by King Mongkut’s Universityof Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. The first authorwas supported by the “Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University ofTechnology Thonburi”.


References

1. Abubakar, A.B.; Kumam, P.; Awwal, A.M. A Descent Dai-Liao Projection Method for Convex ConstrainedNonlinear Monotone Equations with Applications. Thai J. Math. 2018, 17, 128–152.

2. Abubakar, A.B.; Kumam, P. A descent Dai-Liao conjugate gradient method for nonlinear equations.Numer. Algorithms 2019, 81, 197–210. [CrossRef]

3. Abubakar, A.B.; Kumam, P. An improved three-term derivative-free method for solving nonlinear equations.Comput. Appl. Math. 2018, 37, 6760–6773. [CrossRef]

4. Mohammad, H.; Abubakar, A.B. A positive spectral gradient-like method for nonlinear monotone equations.Bull. Comput. Appl. Math. 2017, 5, 99–115.

5. Muhammed, A.A.; Kumam, P.; Abubakar, A.B.; Wakili, A.; Pakkaranang, N. A New Hybrid Spectral GradientProjection Method for Monotone System of Nonlinear Equations with Convex Constraints. Thai J. Math.2018, 16, 125–147.

6. Zhou, W.J.; Li, D.H. A globally convergent BFGS method for nonlinear monotone equations without anymerit functions. Math. Comput. 2008, 77, 2231–2240. [CrossRef]

7. Yan, Q.R.; Peng, X.Z.; Li, D.H. A globally convergent derivative-free method for solving large-scale nonlinearmonotone equations. J. Comput. Appl. Math. 2010, 234, 649–657. [CrossRef]

8. DiRksEandM, S.P.; FERRis, C. A collection of nonlinear mixed complementarity problems. Optim. MethodsSoftw. 1995, 5, 319–345.

9. Meintjes, K.; Morgan, A.P. A methodology for solving chemical equilibrium systems. Appl. Math. Comput.1987, 22, 333–361. [CrossRef]

10. Bellavia, S.; Macconi, M.; Morini, B. STRSCNE: A Scaled Trust-Region Solver for Constrained NonlinearEquations. Comput. Optim. Appl. 2004, 28, 31–50. [CrossRef]

11. Kanzow, C.; Yamashita, N.; Fukushima, M. Levenberg–Marquardt methods with strong local convergenceproperties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 2004, 172, 375–397.[CrossRef]

12. Papp, Z.; Rapajic, S. FR type methods for systems of large-scale nonlinear monotone equations. Appl. Math.Comput. 2015, 269, 816–823. [CrossRef]

329


13. Zhou, W.; Wang, F. A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math.Comput. 2015, 261, 1–7. [CrossRef]

14. Dai, Y.H.; Yuan, Y. A nonlinear conjugate gradient method with a strong global convergence property.SIAM J. Optim. 1999, 10, 177–182. [CrossRef]

15. Polak, E.; Ribiere, G. Note sur la convergence de méthodes de directions conjuguées. Revue FrançaiseD’informatique et de Recherche Opérationnelle Série Rouge 1969, 3, 35–43. [CrossRef]

16. Polyak, B.T. The conjugate gradient method in extremal problems. USSR Comput. Math. Math. Phys. 1969,9, 94–112. [CrossRef]

17. Hager, W.W.; Zhang, H. A new conjugate gradient method with guaranteed descent and an efficient linesearch. SIAM J. Optim. 2005, 16, 170–192. [CrossRef]

18. Dai, Y.H.; Liao, L.Z. New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math.Optim. 2001, 43, 87–101. [CrossRef]

19. Fletcher, R.; Reeves, C.M. Function minimization by conjugate gradients. Comput. J. 1964, 7, 149–154.[CrossRef]

20. Xiao, Y.; Zhu, H. A conjugate gradient method to solve convex constrained monotone equations withapplications in compressive sensing. J. Math. Anal. Appl. 2013, 405, 310–319. [CrossRef]

21. Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations.In Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods; Springer: Boston, MA,USA, 1998; pp. 355–369.

22. Liu, S.Y.; Huang, Y.Y.; Jiao, H.W. Sufficient descent conjugate gradient methods for solving convexconstrained nonlinear monotone equations. In Abstract and Applied Analysis; Hindawi: New York, NY,USA, 2014; Volume 2014.

23. Liu, J.; Li, S. A projection method for convex constrained monotone nonlinear equations with applications.Comput. Math. Appl. 2015, 70, 2442–2453. [CrossRef]

24. Sun, M.; Liu, J. Three derivative-free projection methods for nonlinear equations with convex constraints.J. Appl. Math. Comput. 2015, 47, 265–276. [CrossRef]

25. Sun, M.; Liu, J. New hybrid conjugate gradient projection method for the convex constrained equations.Calcolo 2016, 53, 399–411. [CrossRef]

26. Ou, Y.; Li, J. A new derivative-free SCG-type projection method for nonlinear monotone equations withconvex constraints. J. Appl. Math. Comput. 2018, 56, 195–216. [CrossRef]

27. Ding, Y.; Xiao, Y.; Li, J. A class of conjugate gradient methods for convex constrained monotone equations.Optimization 2017, 66, 2309–2328. [CrossRef]

28. Liu, J.; Feng, Y. A derivative-free iterative method for nonlinear monotone equations with convex constraints.Numer. Algorithms 2018, 1–18. [CrossRef]

29. Kabanikhin, S.I. Definitions and examples of inverse and ill-posed problems. J. Inverse Ill-Posed Probl. 2008,16, 317–357. [CrossRef]

30. Belishev, M.I.; Kurylev, Y.V. Boundary control, wave field continuation and inverse problems for the waveequation. Comput. Math. Appl. 1991, 22, 27–52. [CrossRef]

31. Beilina, L.; Klibanov, M.V. A Globally Convergent Numerical Method for a Coefficient Inverse Problem.SIAM J. Sci. Comput. 2008, 31, 478–509. [CrossRef]

32. Kabanikhin, S.; Shishlenin, M. Boundary control and Gel’fand–Levitan–Krein methods in inverse acousticproblem. J. Inverse Ill-Posed Probl. 2004, 12, 125–144. [CrossRef]

33. Lukyanenko, D.; Grigorev, V.; Volkov, V.; Shishlenin, M. Solving of the coefficient inverse problem fora nonlinear singularly perturbed two dimensional reaction diffusion equation with the location of movingfront data. Comput. Math. Appl. 2019, 77, 1245–1254. [CrossRef]

34. Van, S.B.; Elber, G. Solving piecewise polynomial constraint systems with decomposition and asubdivision-based solver. Computer-Aided Design 2017, 90, 37–47.

35. Aizenshtein, M.; Barton, M.; Elber, G. Global solutions of well-constrained transcendental systems usingexpression trees and a single solution test. Computer Aided Geometric Design 2012, 29, 265–279.

36. Barton, M. Solving polynomial systems using no-root elimination blending schemes. Computer-Aided Design2011, 43, 1870–1878.

37. Wang, X.Y.; Li, S.J.; Kou, X.P. A self-adaptive three-term conjugate gradient method for monotone nonlinearequations with convex constraints. Calcolo 2016, 53, 133–145. [CrossRef]

330


38. La Cruz, W.; Martínez, J.; Raydan, M. Spectral residual method without gradient information for solvinglarge-scale nonlinear systems of equations. Math. Comput. 2006, 75, 1429–1448. [CrossRef]

39. Bing, Y.; Lin, G. An Efficient Implementation of Merrills Method for Sparse or Partially Separable Systems ofNonlinear Equations. SIAM J. Optim. 1991, 1, 206–221, doi:10.1137/0801015. [CrossRef]

40. Yu, Z.; Lin, J.; Sun, J.; Xiao, Y.H.; Liu, L.Y.; Li, Z.H. Spectral gradient projection method for monotonenonlinear equations with convex constraints. Appl. Numer. Math. 2009, 59, 2416–2423. [CrossRef]

41. Dolan, E.D.; Moré, J.J. Benchmarking optimization software with performance profiles. Math. Program. 2002,91, 201–213. [CrossRef]

42. Hale, E.T.; Yin, W.; Zhang, Y. A fixed-point continuation method for �1-regularized minimization withapplications to compressed sensing. CAAM TR07-07 Rice Univ. 2007, 43, 44.

43. Beck, A.; Teboulle, M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J.Imaging Sci. 2009, 2, 183–202. [CrossRef]

44. Figueiredo, M.A.; Nowak, R.D.; Wright, S.J. Gradient projection for sparse reconstruction: Application tocompressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 2007, 1, 586–597. [CrossRef]

45. Birgin, E.G.; Martínez, J.M.; Raydan, M. Nonmonotone spectral projected gradient methods on convex sets.SIAM J. Optim. 2000, 10, 1196–1211. [CrossRef]

46. Xiao, Y.; Wang, Q.; Hu, Q. Non-smooth equations based method for �1-norm problems with applications tocompressed sensing. Nonlinear Anal. Theory Methods Appl. 2011, 74, 3570–3577. [CrossRef]

47. Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility tostructural similarity. IEEE Trans. Image Process. 2004, 13, 600–612. [CrossRef] [PubMed]


331

mathematics

Article

A New Class of Iterative Processes for SolvingNonlinear Systems by Using One DividedDifferences Operator

Alicia Cordero 1 , Cristina Jordán 1 , Esther Sanabria 2 and Juan R. Torregrosa 1,*

1 Multidisciplinary Institute of Mathematics, Universitat Politènica de València, 46022 València, Spain2 Department of Applied Mathematics, Universitat Politènica de València, 46022 València, Spain* Correspondence: [email protected]

Received: 23 July 2019; Accepted: 19 August 2019; Published: 23 August 2019��

Abstract: In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinearsystems is presented. The fourth-order convergence for all the elements of the class is established,proving, in addition, that one element of this family has order five. The proposed methods havefour steps and, in all of them, the same divided difference operator appears. Numerical problems,including systems of academic interest and the system resulting from the discretization of theboundary problem described by Fisher’s equation, are shown to compare the performance ofthe proposed schemes with other known ones. The numerical tests are in concordance with thetheoretical results.

Keywords: nonlinear systems; multipoint iterative methods; divided difference operator; order ofconvergence; Newton’s method; computational efficiency index

1. Introduction

The design of iterative processes for solving scalar equations, f (x) = 0, or nonlinear systems,F(x) = 0, with n unknowns and equations, is an interesting challenge of numerical analysis. Manyproblems in Science and Engineering need the solution of a nonlinear equation or system in any stepof the process. However, in general, both equations and nonlinear systems have no analytical solution,so we must resort to approximate the solution using iterative techniques. There are different ways todevelop iterative schemes such as quadrature formulaes, Adomian polynomials, divided differenceoperator, weight function procedure, etc. have been used by many researchers for designing iterativeschemes to solve nonlinear problems. For a good overview on the procedures and techniques as wellas the different schemes developed in the last half century, one refers to some standard texts [1–5].

In this paper, , we want to design Jacobian-free iterative schemes for approximating the solutionx = (x1, x2, . . . , xn)T of a nonlinear system F(x) = 0, where F : D ⊆ Rn → Rn is a nonlinearmultivariate function defined in a convex set D. The best known method for finding a solution x ∈ Dis Newton’s procedure,

x(k+1) = x(k) − [F′(x(k))]−1F(x(k)), k = 0, 1, 2, . . . ,

F′(x(k)) being the Jacobian of F evaluated in the kth iteration.Based on Newton-type schemes and by using different techniques, several methods for

approximating a solution of F(x) = 0 have been published recently. The main objective of all theseprocesses is to speed the convergence or increase their computational efficiency. We are going to recallsome of them that we will use in the last section for comparison purposes.



From a variant of Steffensen’s method for systems introduced by Samanskii in [6] that replacesthe Jacobian matrix F′(x) by the divided difference operator defined as

[x, y; F](x− y) = F(x)− F(y),

being x, y ∈ Rn, Wang and Fang in [7] designed a fourth-order scheme, denoted by WF4, whoseiterative expression is

r(k) = x(k) − [a(k), b(k); F]−1F(x(k)),

x(k+1) = r(k) −(

3I − 2[a(k), b(k); F]−1[x(k), r(k); F])[a(k), b(k); F]−1F(r(k)),

(1)

where I is the identity matrix of size n × n, a(k) = x(k) + F(x(k)) and b(k) = x(k) − F(x(k)). Let usobserve that this method uses two functional evaluations and two divided difference operators periteration. Let us remark that Samanskii in [6] defined also a third-order method with the same divideddifferences operator at the two steps.

Sharma and Arora in [8] added a new step in the previous method obtaining a sixth-order scheme,denoted by SA6, whose expression is

r(k) = x(k) − [a(k), b(k); F]−1F(x(k)),

s(k) = r(k) −(

3I − 2[a(k), b(k); F]−1[x(k), r(k); F])[a(k), b(k); F]−1F(r(k)),

x(k+1) = s(k) −(

3I − 2[a(k), b(k); F]−1[x(k), r(k); F])[a(k), b(k); F]−1F(s(k)),

(2)

where, as before, a(k) = x(k) + F(x(k)) and b(k) = x(k) − F(x(k)). In relation with WF4, a new functionalevaluation, per iteration, is needed.

By replacing the third step of equation (2), Narang et al. in [9] proposed the followingseventh-order scheme that uses two divided difference operators and three functional evaluations periteration, which is denoted by NM7,

r(k) = x(k) −Q−1F(x(k)),

s(k) = r(k) −Q−1F(r(k)),

x(k+1) = s(k) −(

174

I −Q−1 P(−27

4I + Q−1 P

(194

I − 54

Q−1 P)))

Q−1 F(s(k)),

(3)

where Q = [a(k), b(k); F], being again a(k) = x(k) + F(x(k)), b(k) = x(k) − F(x(k)) and P = [w(k), t(k); F],with w(k) = s(k) + F(x(k)), t(k) = s(k) − F(x(k)).

In a similar way, Wang et al. (see [10]) designed a scheme of order 7 that we denote by S7,modifying only the third step of expression (3). Its iterative expression is

r(k) = x(k) −Q−1F(x(k)),

s(k) = r(k) −(

3I − 2Q−1[r(k), x(k); F])

Q−1F(r(k)),

x(k+1) = s(k) −(

134

I −Q−1 [s(k), r(k); F](

72

I − 54

Q−1 [s(k), r(k); F]))

Q−1 F(s(k)),

(4)

where, as in the previous schemes, Q = [a(k), b(k); F], with a(k) = x(k) + F(x(k)) and b(k) = x(k) − F(x(k)).Different indices can be used to compare the efficiency of iterative processes. For example, in [11],

Ostrowski introduced the efficiency index EI = p1/d, where p is the convergence order and d is thequantity of functional evaluations at each iteration. Moreover, the matrix inversions appearing inthe iterative expressions are in practice calculated by solving linear systems. Therefore, the amountof quotients/products, denoted by op, employed in each iteration play an important role. This is

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the reason why we presented in [12] the computational efficiency index, CEI, combining EI and thenumber of operations per iteration. This index is defined as CEI = p1/(d+op).

Our goal of this manuscript is to construct high-order Jacobian-free iterative schemes for solvingnonlinear systems involving low computational cost on large systems.

We recall, in Section 2, some basic concepts that we will use in the rest of the manuscript. Section 3is devoted to describe our proposed iterative methods for solving nonlinear systems and to analyze theirconvergence. The efficiency indices of our methods are studied in Section 4, as well as a comparativeanalysis with the schemes presented in the Introduction. Several numerical tests are shown in Section 5,for illustrating the performance of the new schemes. To get this aim, we use a discretized nonlinearone-dimensional heat conduction equation by means of approximations of the derivatives and alsosome systems of academic interest. We finish the manuscript with some conclusions.

2. Basic Concepts

If a sequence {x(k)}k≥0 in Rn converges to x, it is said to be of order of convergence p, being p ≥ 1,if C > 0 (0 < C < 1 for p = 1) and k0 exist satisfying

‖x(k+1) − x‖ ≤ C‖x(k) − x‖p, ∀k ≥ k0,

or‖e(k+1)‖ ≤ C‖e(k)‖p, ∀k ≥ k0,

being e(k) = x(k) − x.Although this notation was presented by the authors in [12], we show it for the sake of

completeness. Let Φ : D ⊆ Rn −→ Rn be sufficiently Fréchet differentiable in D. The qth derivativeof Φ at x ∈ Rn, q ≥ 1, is the q-linear function Φ(q)(x) : Rn × · · · × Rn −→ Rn such thatΦ(q)(x)(y1, . . . , yq) ∈ Rn. Let us observe that

1. Φ(q)(x)(y1, . . . , yq−1, ·) ∈ L(Rn), where L(Rn) denotes the set of linear mappings defined from(Rn) into (Rn).

2. Φ(q)(x)(yσ(1), . . . , yσ(q)) = Φ(q)(x)(y1, . . . , yq), for all permutation σ of {1, 2, . . . , q}.

From the above properties, we can use the following notation (let us observe that yp denotes(y, . . . , y) p times):

(a) Φ(q)(x)(y1, . . . , yq) = Φ(q)(x)y1 . . . yq,

(b) Φ(q)(x)yq−1Φ(p)yp = Φ(q)(x)Φ(p)(x)yq+p−1.

Let us consider x + ε ∈ Rn in a neighborhood of x. By applying Taylor series and considering thatΦ′(x) is nonsingular,

Φ(x + ε) = Φ′(x)

[ε +

p−1

∑q=2

Cqεq

]+O(εp), (5)

being Cq = (1/q!)[Φ′(x)]−1Φ(q)(x), q ≥ 2. Let us notice that Cqεq ∈ Rn as Φ(q)(x) ∈ L(Rn × · · · ×Rn,Rn) and [Φ′(x)]−1 ∈ L(Rn).

Moreover, we express Φ′ as

Φ′(x + ε) = Φ′(x)

[I +

p−1

∑q=2

qCqεq−1

]+O(εp−1), (6)

the identity matrix being denoted by I. Then, qCqεq−1 ∈ L(Rn). From expression (6), we get

[Φ′(x + ε)]−1 =[

I + Y2ε + Y3ε2 + Y4ε4 + · · ·][Φ′(x)]−1 +O(εp−1), (7)

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whereY2 = −2C2,Y3 = 4C2

2 − 3C3,Y4 = −8C3

2 + 6C2C3 + 6C3C2 − 4C4....

The equation

e(k+1) = Ke(k)p+O(e(k)

p+1),

where K is a p-linear operator K ∈ L(Rn × · · · ×Rn,Rn), known as error equation and p is the orderof convergence.In addition, we denote e(k)

pby (e(k), e(k), · · · , e(k)).

Divided difference operator of function Φ (see, for example, [2]) is defined as a mapping [·, ·; Φ] :D× D ⊆ Rn ×Rn → L(Rn) satisfying

[x, y; Φ](x− y) = Φ(x)−Φ(y), for all x, y ∈ D.

In addition, by using the formula of Gennochi-Hermite [13] and Taylor series expansions aroundx, the divided difference operator is defined for all x, x + h ∈ Rn as follows:

[x + ε, x; Φ] =∫ 1

0Φ′(x + tε)dt = Φ′(x) +

12

Φ′′(x)ε +16

Φ′′′(x)ε2 +1

24Φ(iv)(x)ε3 +O(ε4). (8)

Being a(k) = x(k) + Φ(x(k)) and b(k) = x(k) −Φ(x(k)), the divided difference operator for pointsa(k) and b(k) is

[a(k), b(k); Φ] = Φ′(b(k)) + 12

Φ′′(b(k))(a(k) − b(k)) +16

Φ′′′(b(k))(a(k) − b(k))2 +O((a(k) − b(k))3) (9)

= Φ′(x)[I + A1e(k) + A2e(k)2+ A3e(k)

3] +O(e(k)

4),

where

A1 = 2C2,

A2 = C3(3 + Φ′(x)2),

A3 = 4C4(1 + Φ′(x)2 + C3Φ′(x)2C2 + C3Φ′(x)C2Φ′(x)

are obtained by replacing the Taylor expansion of the different terms that appear in development (9) and doingalgebraic manipulations.

For computational purposes, the following expression (see [2]) is used

[y, x; F]i,j =Fi(y1, . . . , yj−1, yj, xj+1, . . . , xn)− Fi(y1, . . . , yj−1, xj, xj+1, . . . , xn)

yj − xj,

where x = (x1, . . . , xkj−1, xj, xj+1, . . . , xn) and y = (y1, . . . , yj−1, yj, yj+1, . . . , yn) and 1 ≤ i, j ≤ n.

3. Proposed Methods and Their Convergence

From a Samanskii-type method and by using the composition procedure “frozening” the divided differenceoperator (we hold the same divided difference operator in all the steps of the method), we propose the followingfour-steps iterative class with the aim of reaching order five:

y(k) = x(k) − μ[a(k), b(k); F]−1F(x(k)),

z(k) = y(k) − α [a(k), b(k); F]−1F(y(k)),

t(k) = z(k) − β [a(k), b(k); F]−1F(z(k)),

x(k+1) = t(k) − γ [a(k), b(k); F]−1F(t(k)),

(10)

where μ, α, β and γ are real parameters, a(k) = x(k) + F(x(k)) and b(k) = x(k) − F(x(k)).

335


It is possible to prove that, under some assumptions, we can reach order five. We have used differentcombinations of these steps trying to preserve order 5 and reducing the computational cost. The best result wehave been able to achieve is the following:

y(k) = x(k) − [a(k), b(k); F]−1F(x(k)),

z(k) = y(k) − α [a(k), b(k); F]−1F(y(k)),

t(k) = z(k) − β [a(k), b(k); F]−1F(y(k)),

x(k+1) = z(k) − γ [a(k), b(k); F]−1F(t(k)),

(11)

where α, β and γ are real parameters, a(k) = x(k) + F(x(k)) and b(k) = x(k) − F(x(k)). The convergence of class(11) is presented in the following result.

Theorem 1. Let us assume F : D ⊆ Rn −→ Rn being a differentiable enough operator at each point of the openneighborhood D of the solution x of the system F(x) = 0. Let us suppose that F′(x) is continuous and nonsingular in x andthe initial estimation x(0) is near enough to x. Therefore, sequence {x(k)}k≥0 calculated from expression (11) converges to x

with order 4 if α = 2− γ, β =(γ− 1)2

γand for all γ ∈ R, the error equation being

e(k+1) =(5 γ− 1

γC3

2

)e(k)

4+O(e(k)

5).

In addition, if γ = 15 , the order of convergence is five and the error equation is

e(k+1) =(

14C42 − 2C2C3C2 + 6C3C2

2 − 2C2C3F′(x)2C2 + 2C3F′(x)2C22

)e(k)

5+O(e(k)

6),

where Cj =1j![F′(x)]−1F(j)(x), j = 2, 3, . . .

Proof. By using the Taylor expansion of F(x(k)) and its derivatives around x:

F(x(k)) = F′(x)[e(k) + C2e(k)

2+ C3e(k)

3+ C4e(k)

4+ C5e(k)

5]+O(e(k)

6),

F′(x(k)) = F′(x)[

I + 2C2e(k) + 3C3e(k)2+ 4C4e(k)

3+ 5C5e(k)

4]+O(e(k)

5),

F′′(x(k)) = F′(x)[2C2 + 6C3e(k) + 12C4e(k)

2+ 20C5e(k)

3]+O(e(k)

4),

F′′′(x(k)) = F′(x)[6C3 + 24C4e(k) + 60C5e(k)

2]+O(e(k)

3),

F(iv)(x(k)) = F′(x)[24C4 + 120C5e(k)

]+O(e(k)

2),

F(v)(x(k)) = F′(x)[120C5

]+O(e(k)).

From the above expression, by replacing in the first order divided difference operator [a(k), b(k); F], the valuesa(k) = x(k) + F(x(k)), b(k) = x(k) − F(x(k)), we obtain:[

a(k), b(k); F]= F′(x)

[I + 2C2e(k) +

(3C3 + C3F′(x)2

)e(k)

2

+(

4C4 + 4C4F′(x)2 + C3F′(x)2C2 + C3F′(x)C2F′(x))

e(k)3

+(

5C5 + C5F′(x)4 + 10C5F′(x)2 + 4C4F′(x)2C2 + 4C4F′(x)C2F′(x)

+ C3F′(x)2C3 + C3F′(x)C2F′(x)C2 + C3F′(x)C3F′(x))

e(k)4]

+O(e(k)5).

From the above expression, we have[a(k), b(k); F

]−1=[

I + X2e(k) + X3e(k)2][F′(x)]−1 +O(e(k)

3),

336


where

X2 = −2C2,

X3 = −3C3 + 4C22 − C3F′(x)2,

X4 = −4C4 + 6C2C3 + 6C3C2 − 8C32 + C3F′(x)2C2 − C3F′(x)C2F′(x) + 2C2C3F′(x)2 − 4C4F′(x)2.

Then,[a(k), b(k); F

]−1F(x(k)) = e(k) − C2e(k)

2+(− 2C3 + 2C2

2 − C3F′(x)2)

e(k)3

+(− 3C4 + 4C2C3 + 3C3C2 − 4C3

2 − C3F′(x)C2F′(x) + 2C2C3F′(x)2 − 4C4F′(x)2)

e(k)4+O(e(k)

5).

Thus,

y(k) − x = C2e(k)2+(

2C3 − 2C22 + C3F′(x)2

)e(k)

3

+(

3C4 − 4C2C3 − 3C3C2 + 4C32 + C3F′(x)C2F′(x)− 2C2C3F′(x)2 + 4C4F′(x)2

)e(k)

4+O(e(k)

5),

(y(k) − x)2 = C22e(k)

4+O(e(k)

5),

and

F(y(k)) = F′(x)[(y(k) − x) + C2(y(k) − x)2

]+O((y(k) − x)3)

= F′(x)[C2e(k)

2+(

2C3 − 2C22 + C3F′(x)2

)e(k)

3

+(


)e(k)

4]+O(e(k)

5).

From the values of z(k) and t(k) in expression (11), we have

t(k) = y(k) − (α + β) [a(k), b(k); F]−1F(y(k)).

Then,[a(k), b(k); F

]−1F(y(k)) = C2e(k)

2+(

2C3 − 4C22 + C3F′(x)2

)e(k)

3

+(

3C4 − 8C2C3 − 6C3C2 + 13C32

+C3F′(x)C2F′(x)− 4C2C3F′(x)2 + 4C4F′(x)2 − C3F′(x)2C2

)e(k)

4+O(e(k)

5).

Similarly, we obtain

t(k) − x =(

1− (α + β))

C2e(k)2+((

1− (α + β))(

2C3 − 2C22 + C3F′(x)2

)+ 2(α + β)C2

2

)e(k)

3

+((

1− (α + β))(


)−(α + β)

(− 4C2C3 − 3C3C2 + 9C3

2 − 2C2C3F′(x)2 − C3F′(x)2C2

)e(k)

4+O(e(k)

5),

(t(k) − x)2 =(

1− (α + β))2

C22e(k)

4+O(e(k)

5)

and

F(t(k)) = F′(x)[(

1− (α + β))

C2e(k)2+((

1− (α + β))(

2C3 − 2C22 + C3F′(x)2

)+ 2(α + β)C2

2

)e(k)

3

+((

1− (α + β))(


)−(α + β)

(− 4C2C3 − 3C3C2 + 9C3

2 − 2C2C3F′(x)2 − C3F′(x)2C2

)+(

1− (α + β))2

C32

)e(k)

4]+O(e(k)

5).

337


Thus,[a(k), b(k); F

]−1F(t(k)) =

(1− (α + β)

)C2e(k)

2

+((

1− (α + β))(

2C3 − 4C22 + C3F′(x)2

)+ 2(α + β)C2

2

)e(k)

3

+((

1− (α + β))(

3C4 − 8C2C3 − 6C3C2 + 12C32 + C3F′(x)C2F′(x)− 4C2C3F′(x)2

−C3F′(x)2C2 + 4C4F′(x)2)+(

1− (α + β))2

C32

−(α + β)(− 4C2C3 − 3C3C2 + 13C3

2 − C3F′(x)2C2 − 2C2C3F′(x)2))

e(k)4+O(e(k)

5).

Therefore, we obtain

e(k+1) = e(k) −[

a(k), b(k); F]−1

F(x(k))− α[

a(k), b(k); F]−1

F(y(k))− γ[

a(k), b(k); F]−1

F(t(k))

=(

1− α− γ(

1− (α + β)))

C2e(k)2+[(

1− α− γ(

1− (α + β)))(

2C3 − 2C22 + C3F′(x)2

)− 2

(− α + γ(α + β)− γ

(1− (α + β)

))C2

2

]e(k)

3

+[(

1− α− γ(

1− (α + β)))(


)+(− α + γ(α + β)− γ

(1− (α + β)

))(− 4C2C3 − 3C3C2 + 8C3

2 − 2C2C3F′(x)2 − C3F′(x)2C2

)(− α + 5γ(α + β)− γ

(1− (α + β)

)2)C3

2

]e(k)

4+O(e(k)

5).

Thus, by requiring that the coefficients of e(k)2

and e(k)3

are null, we get α = 2− γ and β =(γ− 1)2

γfor all

γ ∈ R, 4 being its order,

e(k+1) =(5 γ− 1

γC3

2

)e(k)

4+O(e(k)

5).

By adding the coefficient of e(k)4

to the above system, we get γ = 15 , 5 being the order of convergence with

error equation in this case:

e(k+1) =(

14C42 − 2C2C3C2 + 6C3C2

2 − 2C2C3F′(x)2C2 + 2C3F′(x)2C22

)e(k)

5+O(e(k)

6).

4. Efficiency Indices

As we have mentioned in the Introduction, we use indices EI = p1/d and CEI to compare the differentiterative methods.

To evaluate function F, n scalar functions are calculated and n(n− 1) for the first order divided difference[·, ·; F]. In addition, to calculate an inverse linear operator, an n× n linear system must be solved; then, we have to

do13

n3 + n2 − 13

n quotients/products for getting LU decomposition and solving the corresponding triangular

linear systems. Moreover, for solving m linear systems with the same matrix of coefficients, we need to do13

n3 + mn2 − 13

n products-quotients. In addition, we need n2 products for each matrix-vector multiplication and

n2 quotients for evaluating a divided difference operator.According to the last considerations, we calculate the efficiency indices EI of methods CJST5, NM7, S7, SA6

and WF4. In case CJST5, for each iteration, we evaluate F three times and once [·, ·; F], so n2 + 2n functional

evaluations are needed. Therefore, EICJST5 = 51

n2+2n . The indices obtained for the mentioned methods are alsocalculated and shown in Table 1.

338


Table 1. Efficiency indices for different methods.

Method Order NFE EI

NM7 7 2n2 + n 71

2n2+n

S7 7 3n2 71

3n2

SA6 6 2n2 + n 61

2n2+n

WF4 4 2n2 41

2n2

CJST5 5 n2 + 2n 51

n2+2n

In Table 2, we present the indices CEI of schemes NM7, SA6, S7, WF4 and CJST5. In it, the amount offunctional evaluations is denoted by NFE, the number of linear systems with the same [·, ·; F] as the matrixof coefficients is NLS1 and M× V represents the quantity of products matrix-vector. Then, in case CJST5, foreach iteration, n2 + 2n functional evaluations are needed, since we evaluate three times the function F and onedivided difference of first order [a(k), b(k), F]. In addition, we must solve three linear systems with [a(k), b(k), F] as

coefficients matrix (that is13

n3 + 3n2 − 13

n). Thus, the value of CEI for CJST5 is

CEICJST5 = 51

13 n3+5n2+ 5

3 n .

Analogously, we obtain the indices CEI of the other methods. In Figure 1, we observe the computationalefficiency index of the different methods of size 5 to 80. The best index corresponds to our proposed scheme.

1 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1.009

1 . 01

5 10 15 20

NM7 S7 SA6 WF4 CJTS5

(a) Size from 5 to 20

1

1.00002

1.00004

1.00006

1.00008

1.0001

1.00012

1.00014

30 40 50 60 70 80

NM7 S7 SA6 WF4 CJTS5

(b) Size from 30 to 80

Figure 1. CEI for several sizes of the system.

Table 2. Computational cost of the procedures.

Method Order NFE NLS1 M × V CEI

NM7 7 2n2 + 3n 4 3 71

13 n3+11n2+ 2

3 n

S7 7 3n2 6 3 71

13 n3+14n2− 1

3 n

SA6 6 2n2 + n 4 2 61

13 n3+11n2+ 2

3 n

WF4 4 2n2 3 1 41

13 n3+8n2− 1

3 n

CJST5 5 n2 + 2n 3 0 51

13 n3+5n2+ 5

3 n


We begin this section checking the performance of the new method on the resulting system obtained bythe discretization of Fisher’s partial differential equation. Thereafter, we compare its behavior with that of otherknown methods on some academic problems. For the computations, we have used Matlab R2015a ( Natick,Massachusetts, USA) with variable precision arithmetic, with 1000 digits of mantissa. The characteristics of the

339


computer are, regarding the processor, Intel(R) Core(TM) i7-7700 CPU @ 3.6 GHz, 3.601 Mhz, four processors andRAM 16 GB.

We use the estimation of the theoretical order of convergence p, called Computational Order of Convergence(COC), introduced by Jay [14] with the following expression:

p ≈ COC =ln(‖F(x(k+1))‖2/‖F(x(k))‖2

)ln (‖F(x(k))‖2/‖F(x(k−1))‖2)

, k = 1, 2, . . .

and the Approximated Computational Order of Convergence (ACOC), defined by Cordero and Torregrosa in [15]

p ≈ ACOC =ln (‖x(k+1) − x(k)‖2/‖x(k) − x(k−1)‖2)

ln (‖x(k) − x(k−1)‖2/‖x(k−1) − x(k−2)‖2).

Example 1. Fisher’s equation,

ut = Duxx + ru(

1− up

), x ∈ [a, b] t ≥ 0, (12)

was proposed in [16] by Fisher to model the diffusion process in population dynamics. In it, D > 0 is the diffusion constant,r is the level of growth of the species and p is the carrying capacity. Lately, this formulation has proven to be fruitful formany other problems as wave genetics, economy or propagation.

Now, we study a particular case of this equation, when r = p = 1 and the spatial interval is [0, 1], u(x, 0) =sech2(πx) and null boundary conditions.

We transform Example 1 in a set of nonlinear systems by applying an implicit method of finite differences,providing the estimated solution in the instant tk from the estimated one in tk−1. The spacial step h = 1/nx isselected and the temporal step is k = Tmax/nt, nx and nt being the quantity of subintervals in x and t, respectively,and Tmax is the final instant. Therefore, a grid of domain [0, 1]× [0, Tmax] with points (xi, tj), is selected:

xi = 0 + ih, i = 0, 1, . . . , nx, tj = 0 + jk, j = 0, 1, . . . , nt.

Our purpose is to estimate the solution of problem (12) at these points, by solving many nonlinear systems,as much as the number of temporal nodes tj. For it, we use the following finite differences of order O(k + h2):

ut(x, t) ≈ u(x, t)− u(x, t− k)k

,

uxx(x, t) ≈ u(x + h, t)− 2u(x, t) + u(x− h, t)h2 .

By denoting the approximation of the solution at (xi, tj) as ui,j, and, by replacing it in Example 1, we getthe system

kui+1,j + (kh2 − 2k− h2)ui,j − kh2u2i,j + kui−1,j = −h2ui,j−1,

for i = 1, 2, . . . , nx − 1 and j = 1, 2, . . . , nt. The unknowns of this system are u1,j, u2,j, . . . , unx−1,j, that is,the approximations of the solution in each spatial node for the fixed instant tj. Let us remark that, for solving thissystem, the knowledge of solution in tj−1 is required.

Let us observe (Table 3) that the results improve when the temporal step is smaller. In this case, the COC isnot a good estimation of the theoretical error. In Figure 2, we show the approximated solution of the problemwhen Tmax = 10, by taking nt = 50, nx = 10 and using method CJST5.

Table 3. Fisher results by CJST5 and different Tmax.

Tmax nx nt ‖F(x1)‖ ‖F(x2)‖ ‖F(x3)‖ COC

0.1 10 20 8.033 × 10−9 2.356 × 10−35 3.243 × 10−68 1.23850.1 10 200 8.679 × 10−13 3.203 × 10−44 7.623 × 10−77 1.0379

1 10 20 4.158 × 10−5 3.4 × 10−25 1.679 × 10−56 1.55851 10 200 8.033 × 10−9 2.356 × 10−35 3.243 × 10−68 1.2385

10 10 20 nc10 10 50 0.01945 2.757 × 10−11 1.953 × 10−38 3.0683

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Figure 2. Approximated solution of Example 1.

In the rest of examples, we are going to compare the performance of the proposed method with the schemespresented in the Introduction as well as with the Newton-type method replacing the Jacobian matrix by thedivided difference operator, that is, the Samanskii’s scheme (see [6]).

Example 2. Let us define the nonlinear system⎧⎪⎨⎪⎩cos(x2)− sin(x1) = 0,xx1

3 − 1/x2 = 0,ex1 − x2

3 = 0.

We use, in this example, the starting estimation x(0) = (1.25, 1.25, 1.25)T , the solution being x ≈(0.9096, 0.6612, 1.576)T . Table 4 shows the residuals ‖x(k) − x(k−1)‖ and ‖F(x(k))‖ for k = 1, 2, 3 as well asACOC and COC. We observe that the COC index is better than the corresponding ACOC of the other methods.In addition, the value ‖x3 − x2‖ is better or similar to that of S7 and NM7 methods, both of them of order 7.

Table 4. Numerical results for Example 2.

Samanskii CJST5 WF4 SA6 S7 NM7

‖x(1) − x(0)‖ 1.415 0.8848 0.8539 0.8934 0.9148 0.9355‖x(2) − x(1)‖ 0.5427 0.247 0.2039 0.2689 0.2942 0.3098‖x(3) − x(2)‖ 0.1738 0.01159 0.006249 0.005301 0.01069 0.02667

ACOC 1.1875 2.3976 2.433 3.2705 2.9221 4.25‖F(x(1))‖ 0.1954 0.1282 0.1038 0.1385 0.1817 0.2098‖F(x(2))‖ 0.02369 0.009956 0.004815 0.003584 0.006462 0.01669‖F(x(3))‖ 0.00269 2.805 × 10−8 7.663 × 10−8 4.009 × 10−11 8.074 × 10−12 1.333 × 10−9

COC 1.0313 5.0015 3.5983 5.0104 6.1445 6.4561

Example 3. We consider nown

∑j=1

xj − xi − e−xi xi = 0, i = 1, 2, . . . , n.

The numerical results are displayed in Table 5. The initial estimation is x(0) = (0.25, 0.25, . . . , 0.25)T andthe size of the system is n = 10, the solution being x = (0, 0, . . . , 0)T . We show the same information as in theprevious example.

Example 4. The third example is given by the system:

xi + 1− 2 log

⎛⎝1 +n

∑j=1

xj − xi

⎞⎠ = 0, i = 1, 2, . . . , n.

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Its solution is x ≈ (9.376, 9.376, . . . , 9.376)T . By using the starting guess x(0) = (1, 1, . . . , 1)T with n = 10, weobtain the results appearing in Table 6.



‖x(1) − x(0)‖ 1.036 0.8249 0.9116 0.8499 0.7847 0.7897‖x(2) − x(1)‖ 0.2552 0.03432 0.121 0.05932 0.00583 0.0008995‖x(3) − x(2)‖ 0,009667 1.487 × 10−11 9.264 × 10−6 5.367 × 10−10 2.529 × 10−21 1.048 × 10−28

ACOC 2.3361 6.7807 4.6937 6.9572 8.6247 4.25‖F(x(1))‖ 1.944 0.2742 0.9634 0.4735 0.04665 0.007196‖F(x(2))‖ 0.0773 1.19 × 10−10 7.411 × 10−5 4.293 × 10−9 2.023 × 10−20 8.381 × 10−28

‖F(x(3))‖ 2.65 × 10−5 2.839 × 10−59 9.644 × 10−22 1.3 × 10−57 9.093 × 10−149 2.621 × 10−202

COC 2.4743 5.1932 4.1045 6.0328 6.9895 6.9987



‖x(1) − x(0)‖ 6.013 40.68 67.88 73.31 34.3 39.23‖x(2) − x(1)‖ 12.15 13.83 36.07 56.74 17.15 7.369‖x(3) − x(2)‖ 10.11 0.0002872 0.06643 0.02688 0.0001485 5.422 × 10−8

ACOC - 9.9941 9.9587 29.874 16.819 4.25‖F(x(1))‖ 10.84 11.03 30.43 48.74 13.42 5.842‖F(x(2))‖ 6.084 0.0002263 0.05234 0.02117 0.000117 4.272 × 10−8

‖F(x(3))‖ 1.162 9.286 × 10−28 6.508 × 10−12 1.807 × 10−20 3.105 × 10−40 7.552 × 10−65

COC 2.8651 4.9888 3.583 5.3744 7.0313 6.9755

The different methods give us the expected results, according to their order of convergence.

Example 5. Finally, the last example that we consider is:

arctan (xi) + 1− 2

⎛⎝ n

∑j=1

x2j − x2

i

⎞⎠ = 0.

The solution of it is x ≈ (0.1758, 0.1758, . . . , 0.1758)T . By using the initial estimation x(0) = (0.5, 0.5, . . . , 0.5)T

with n = 20, we obtain the numerical results displayed in Table 7.



‖x(1) − x(0)‖ 0.9503 1.323 1.272 1.368 1.394 1.393‖x(2) − x(1)‖ 0.3912 0.1266 0.177 0.0821 0.05639 0.05732‖x(3) − x(2)‖ 0.1013 4.988 × 10−5 0.0007407 6.903 × 10−7 7.214 × 10−9 6.655 × 10−9

ACOC 1.5229 3.3404 2.776 4.1543 4.9485 4.25‖F(x(1))‖ 8.324 1.706 2.471 1.075 0.7257 0.7381‖F(x(2))‖ 1.445 0.0006179 0.009181 8.552 × 10−6 8.937 × 10−8 8.245 × 10−8

‖F(x(3))‖ 0.0902 1.206 × 10−20 5.635 × 10−12 5.437 × 10−36 8.115 × 10−56 3.521 × 10−56

COC 1.5839 4.8559 3.791 5.9219 6.953 6.9577

It is observed in Tables 5–7 that, for the proposed academic problems, the introduced method (CJST5) showsa good performance comparable with higher-order methods. Of course, the worst results are those obtained bySamanskii’s method, but it has been included because it is the Jacobian-free version of Newton’s scheme and itis also the first step of our proposed scheme. Let us also remark that, when only three iterations are calculated,the index COC gives more reliable information than the ACOC one in all of the examples.

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6. Conclusions

In this paper, we design a family of iterative methods for solving nonlinear systems with fourth-orderconvergence. This family does not use Jacobian matrices and one of its elements has order five. The relationshipbetween the proposed method and other known ones in terms of efficiency index and computational efficiencyindex allows us to see that our method is more efficient than the other ones. In addition, its error bounds aresmaller with the same number of iterations in some cases. Thus, our proposal is competitive mostly for bigsize systems.

Author Contributions: The individual contributions of the authors are as follows: conceptualization, J.R.T.;writing—original draft preparation, C.J. and E.S.; validation, A.C. and J.R.T.; formal analysis, A.C.; numericalexperiments, C.J. and E.S.

Funding: This research has been supported partially by Spanish Ministerio de Ciencia, Innovación yUniversidades PGC2018-095896-B-C22, PGC2018-094889-B-I00, TEC2016-79884-C2-2-R and also by Spanish grantPROMETEO/2016/089 from Generalitat Valenciana.

Acknowledgments: The authors would like to thank the anonymous reviewers for their useful comments andsuggestions that have improved the final version of this manuscript.

Conflicts of Interest: The authors declare that there is no conflict of interest regarding the publication of this paper.

References

1. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.2. Ortega, J.M.; Rheinbolt, W.C. Iterative Solutions of Nonlinears Equations in Several Variables; Academic Press:

New York, NY, USA, 1970.3. Kelley, C.T. Iterative Methods for Linear and Nonlinear Equations; SIAM: Philadelphia, PA, USA, 1995.4. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Dzunic, J. Multipoint Methods for Solving Nonlinear Equations;

Academic Press: New York, NY, USA, 2013.5. Amat, S.; Busquier, S. Advances in Iterative Methods for Nonlinear Equations; SEMA SIMAI Springer Series;

Springer International Publishing: Cham, Switzerland, 2016; Volume 10.6. Samanskii, V. On a modification of the Newton method. Ukrain. Mat. 1967, 19, 133–138.7. Wang, X.; Fang, X. Two Efficient Derivative-Free Iterative Method for Solving Nonlinear Systems. Algorithms

2016, 9, 14. [CrossRef]8. Sharma, J.R.; Arora, H. Efficient derivative-free numerical methods for solving systems of nonlinear

equations. Comp. Appl. Math. 2016, 35, 269–284. [CrossRef]9. Narang, M.; Bathia S.; Kanwar, V. New efficient derivative free family of seventh-order methods for solving

systems of nonlinear equations. Numer. Algorithms 2017, 76, 283–307. [CrossRef]10. Wang, X.; Zhang, T.; Qian, W.; Teng, M. Seventh-order derivative-free iterative method for solving nonlinear

systems. Numer. Algorithms 2015, 70, 545–558. [CrossRef]11. Ostrowski, A.M. Solution of Equations and Systems of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.12. Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. A modified Newton-Jarratt’s composition. Numer.

Algorithms 2010, 55, 87–99. [CrossRef]13. Hermite, C. Sur la formule d’interpolation de Lagrange. Reine Angew. Math. 1878, 84, 70–79. [CrossRef]14. Jay, L.O. A note of Q-order of convergence. BIT Numer. Math. 2001, 41, 422–429. [CrossRef]15. Cordero, A; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math.

Comput. 2007, 190, 686–698. [CrossRef]16. Fisher, R.A. The wave of advance of advantageous genes. Ann. Eugen. 1937, 7, 353–369. [CrossRef]


343

mathematics

Article

An Efficient Iterative Method Based on Two-StageSplitting Methods to Solve WeaklyNonlinear Systems

Abdolreza Amiri 1, Mohammad Taghi Darvishi 1,* , Alicia Cordero 2 and

Juan Ramón Torregrosa 2

1 Department of Mathematics, Razi University, Kermanshah 67149, Iran2 Institute for Multidisciplinary Mathematics, Universitat Politècnica de València,

Camino de Vera s/n, 46022 Valencia, Spain* Correspondence: [email protected]; Tel.: +98-83-3428-3929

Received: 17 July 2019; Accepted: 19 August 2019; Published: 3 September 2019

Abstract: In this paper, an iterative method for solving large, sparse systems of weakly nonlinearequations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme.Under suitable assumptions, we establish the convergence theorem for this method. In addition,it is shown that any faster and less time-consuming two-stage splitting method that satisfies theconvergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such asCPU time, show the robustness of our new method. This method is easy, fast and convenient with anaccurate solution.

Keywords: system of nonlinear equations; Newton method; Newton-HSS method; nonlinearHSS-like method; Picard-HSS method

1. Introduction

For G : D ⊆ Cm −→ Cm, we consider the following system of nonlinear equations:

G(x) = 0. (1)

One may encounter equations like (1) in some areas of scientific computing. In particular,when the technique of finite elements or finite differences are used to discretize nonlinear boundaryproblems, integral equations and certain nonlinear partial differential equations. Finding the roots ofsystems like (1) has widespread applications in numerical and applied mathematics. There are manyiterative schemes to solve (1). The most common one is the second order classical Newton’s scheme,which solves (1) iteratively as

x(n+1) = x(n) − G′(x(n))−1G(x(n)), n = 0, 1, . . . , (2)

where G′(x(n)) is the Jacobian matrix of G, evaluated in the nth iteration. To avoid computation ofinverse of the Jacobian matrix G′(x), Equation (2) is changed to

G′(x(n))(x(n+1) − x(n)) = −G(x(n)). (3)

Equation (3) is a system of linear equations. Hence, by s(n) = x(n+1) − x(n), we have to solve thefollowing system of equations:

G′(x(n))s(n) = −G(x(n)), (4)



whence x(n+1) = x(n) + s(n). Thus, by using this approach, we have to solve a system of linearequations such as

Ax = b, (5)

which we usually use an iterative scheme to solve it.Furthermore, an inexact Newton method [1–4] is a generalization of Newton’s method for

solving (1), in which, at the nth iteration, the step-size s(n) from current approximate solution x(n)

must satisfy a condition such as

‖ G(x(n)) + G′(x(n))s(n) ‖≤ ηn ‖ G(x(n)) ‖,

for a “forcing term” ηn ∈ [0, 1). Let us consider system (1) in which G(x) can be separated into linearand nonlinear terms, Ax and ϕ(x), respectively, that is

G(x) = ϕ(x)− Ax or Ax = ϕ(x). (6)

In (6), the m×m complex matrix A is a positive definite, large and sparse matrix. In addition,vector-valued function ϕ : D ⊆ Cm −→ Cm is continuously differentiable. Furthermore, x is anm-vector and D is an open set. When the norm of linear part Ax is strongly dominant over the normof nonlinear part ϕ(x) in a specific norm, system (6) is called a weakly nonlinear system [5,6]. Bai [5]used the separability and strong dominance between the linear and the nonlinear parts and introducedthe following iterative scheme

Ax(n+1) = ϕ(x(n)). (7)

Equation (7) is a system of linear equations. When the matrix A is positive definite,Axelsson et al. [7] solved it by a class of nested iteration methods. To solve linear positive definitesystems, Bai et al. [8] applied the Hermitian/skew-Hermitian splitting (HSS) iterative scheme.For solving the large sparse, non-Hermitian positive definite linear systems, Li et al. [9] usedan asymmetric Hermitian/skew-Hermitian (AHSS) iterative scheme. Moreover, to improve therobustness of the HSS method, some HSS-based iterative algorithms have been introduced. Bai andYang [10] presented Picard-HSS and HSS-like methods to solve (7), when matrix A is a positive definitematrix. Based on the matrix multi-splitting technique, block and asynchronous two-stage methods areintroduced by Bai et al. [11]. The Picard circulant and skew-circulant splitting (Picard-CSCS) algorithmand the nonlinear CSCS-like iterative algorithm are presented by Zhu and Zhang [12], when thecoefficient matrix A is a Teoplitz matrix. A class of lopsided Hermitian/skew-Hermitian splitting(LHSS) algorithms and a class of nonlinear LHSS-like algorithms are used by Zhu [6] to solve the largeand sparse of weakly nonlinear systems.

It must be noted that system (6) is a special form of system (1). Generally, system (6) is nonlinear.If we classify Picard-HSS and nonlinear HSS-like iterative methods as Jacobian-free schemes, in manycases, they are not as successful as Jacobian dependent schemes such as the Newton method. Most ofthe methods for solving nonlinear systems need to compute or approximate the Jacobian matrix inthe obtained points at each step of the iterative methods, which is a very time-consuming process,especially when the Jacobian matrices ϕ′(x(n)) are dense. Therefore, introducing any scheme that doesnot need to compute the Jacobian matrix and can solve a wider range of problems than the existingones is welcome. In fact, Jacobian-free methods to solve nonlinear systems are very important andform an attractive area of research.

In this paper, we present a new iterative method to solve weakly nonlinear systems. Even thoughthe new algorithm uses some notions of mentioned algorithms, but differs from all of them because ithas three important characteristics. At the first, the new algorithm is a fully Jacobian-free one. At thesecond, it is easy to use, and, finally, it is very successful to solve weakly nonlinear systems. The new

345


iterative method is a synergistic combination of high order Newton-like methods and a special splittingof the coefficient matrix A in (5).

The rest of this paper has organized as follows: in the following section, we present our newalgorithm. We prove convergence of our algorithm in Section 3. We apply our algorithm to solve someproblems in Section 4. In Section 5, we conclude our results and give some comments and discussions.

2. The New Algorithm

In linear system Ax = b, we suppose that A = H + S, where H = 12 (A + A∗), S = 1

2 (A− A∗),and A∗ is the conjugate transpose of matrix A. Hence, H and S are, respectively, Hermitian andskew-Hermitian parts of A. By an initial guess x0 ∈ Cn, and positive constants α and tol, in HSSscheme [8], one computes xl for l = 1, 2, . . . as{

(αI + H)xl+ 12= (αI − S)xl + b,

(αI + S)xl+1 = (αI − H)xl+ 12+ b,

(8)

where I is the identity matrix. Stopping criterion for (8) is ‖b− Axl‖ ≤ tol‖b− Ax0‖, for known x0

and tol.Bai and Guo [13] used an HSS scheme as inner iterations to generate an inexact version of

Newton’s method as:

(1) Consider the initial guess x(0), α, tol and the sequence {ln}∞n=0 of positive integers.

(2) For n = 1, 2, ... until ‖G(x(n))‖ ≤ tol‖G(x(0))‖ do:

(2.1) Set s(n)0 = 0.(2.2) For l = 1, 2, . . . , ln − 1, apply Algorithm HSS as⎧⎨⎩ (αI + H(x(n)))s(n)

l+ 12= (αI − S(x(n)))s(n)l − G(x(n))

(αI + S(x(n)))s(n)l+1 = (αI − H(x(n)))s(n)l+ 1

2− G(x(n)),

and obtain s(n)lnsuch that

‖ G(x(n)) + G′(x(n))s(n)ln‖≤ ηn ‖ G(x(n)) ‖, for some ηn ∈ [0, 1).

(2.3) Set x(n+1) = x(n) + s(n)ln.

In addition, to solve weakly nonlinear problems, one can use a Picard-HSS method as a simpleand Jacobian-free method, which is described as follows [10].

2.1. Picard-HSS Iteration Method

Suppose that ϕ : D ⊂ Cn → Cn is a continuous function and A ∈ Cn×n is a positive definitematrix. For an initial guess x(0) and for a positive integer sequence {ln}∞

n=0, Picard-HSS iterativemethod computes x(n+1) for n = 0, 1, 2, . . ., by using the following iterative scheme, until the stoppingcriterion is satisfied [10],

(1) Set x(n)l := x(n);(2) For l = 0, 1, 2, . . . , n− 1, obtain x(n+1) from solving the following:⎧⎨⎩ (αI + H)x(n)

l+ 12

= (αI − S)x(n)l + ϕ(x(n)),

(αI + S)x(n)l+1 = (αI − H)x(n)l+ 1

2+ ϕ(x(n)).

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(3) Set x(n+1) := x(n)ln.

The numbers ln, n = 0, 1, 2, . . . depend on the problem, so practically they are difficult to bedetermined in real computations. A modified form of Picard-HSS iteration scheme, called the nonlinearHSS-like method, has been presented [10] to avoid using inner iterations as follows.

2.2. Nonlinear HSS-Like Iteration Method

Obtain x(n+1), n = 0, 1, 2, . . . from the following [10], for a given x(0) ∈ D ⊂ Cn, until the stoppingcondition is satisfied{

(αI + H)x(n+12 ) = (αI − S)x(n) + ϕ(x(n)),

(αI + S)x(n+1) = (αI − H)x(n+12 ) + ϕ(x(n+

12 )).

However, in this method, it is necessary to evaluate the nonlinear term ϕ(x) at each step, whichfor complicated nonlinear terms ϕ(x) is too costly.

2.3. Our Proposal Iterative Scheme

For solving (6) without computing Jacobian matrices, we present a new algorithm. This algorithmis a strong tool for solving weakly nonlinear problems, as Picard and nonlinear Picard algorithms,but, in comparison with Picard and nonlinear Picard algorithms, it solves a wider range of nonlinearsystems. First, we change (7) as

Ax(n+1) = Ax(n) − Ax(n) + ϕ(x(n)) (9)

and

Ax(n+1) − Ax(n) = −Ax(n) + ϕ(x(n)). (10)

After computing x(n), set b(n) = ϕ(x(n)), Gn(x) = b(n) − Ax. Then, by intermediate iterations,obtain x(n+1) as:

• Let x(n)0 = x(n) and until ‖G(x(n)k ) ‖≤ toln‖G(x(n)0 )‖ do:

As(n)k = G(x(n)k ), (11)

where s(n)k = x(n)k+1 − x(n)k (k is the counter of the number of iterations (11)).• For solving (11), one may use any inner solver; here, we use an HSS scheme. Next, for initial value

x(n)0 and k = 1, 2, . . . , kn − 1 until

‖Gn(x(n)k ) ‖≤ toln‖Gn(x(n)0 )‖, (12)

apply the HSS scheme as:

(1) Set s(n)k,0 = 0.(2) For l = 0, 1, 2, . . . , lkn − 1, apply algorithm HSS (l is the counter of the number of HSS iterations):⎧⎨⎩ (αI + H)s(n)

k,l+ 12

= (αI − S)s(n)k,l + Gn(x(n)k ),

(αI + S)s(n)k,l+1 = (αI − H)s(n)k,l+ 1

2+ Gn(x(n)k )

(13)

347


and obtain s(n)k,lknsuch that

‖ Gn(x(n)k )− As(n)k,lkn‖≤ η

(n)k ‖Gn(x(n)k )‖, η

(n)k ∈ [0, 1). (14)

(3) Set x(n)k+1 = x(n)k + s(n)k,lkn(lkn is the required number of HSS inner iterations for satisfying (14)).

• Finally, set x(n+1)0 = x(n)kn

(kn is the required number of iterations (11) in the nth step, for

satisfying (12)), b(n+1) = ϕ(x(n+1)0 ), Gn+1(x) = b(n+1) − Ax and again apply steps 3–14 in

Algorithm 1 until to achieve the following stopping criterion:

‖Ax(n) − ϕ(x(n))‖ ≤ tol ‖ Ax(0) − ϕ(x(0)) ‖ .

Algorithm 1: JFHSS Algorithm

Input: x(0), tol, α, n ← 1Output: The root of Ax− ϕ(x) = 0

1 root ← x(0)

2 while ‖Ax(n) − ϕ(x(n))‖ > tol ‖ Ax(0) − ϕ(x(0)) ‖ doInput: toln

Set: x(n)0 = x(n), b(n) = ϕ(x(n)) and Gn(x) = b(n) − Ax, k = 1.

3 while ‖Gn(x(n)k ) ‖> toln‖Gn(x(n)0 )‖ do

4 Set: l = 0, s(n)k,0 = 0.

5 while ‖ Gn(x(n)k )− As(n)k,lkn‖> η

(n)k ‖Gn(x(n)k )‖ do

6

(the HSS algorithm)

(αI + H)s(n)k,l+ 1

2= (αI − S)s(n)k,l + Gn(x(n)k ),

(αI + S)s(n)k,l+1 = (αI − H)s(n)k,l+ 1

2+ Gn(x(n)k ),

7 if ‖ Gn(x(n)k )− As(n)k,lkn‖≤ η

(n)k ‖Gn(x(n)k )‖ then

8 l ← lkn , x(n)k+1 = x(n)k + s(n)k,lkn

9 else10 l ← l + 1

11 if ‖Gn(x(n)k ) ‖≤ toln‖Gn(x(n)0 )‖ then

12 k ← kn, x(n+1)0 = x(n)kn

13 else14 k ← k + 1

15 if ‖Ax(n) − ϕ(x(n))‖ ≤ tol ‖ Ax(0) − ϕ(x(0)) ‖ then

16 root ← x(n)

17 else

18 n ← n + 1, x(n) = x(n+1)0 , b(n) = ϕ(x(n)), Gn(x) = b(n) − Ax

19 return root

We call this new method a JFHSS (Jacobian-free HSS) algorithm, and its steps are shown inAlgorithm 1.

348


In addition, we call the intermediate iterations Newton-like iteration because this kind of iterationuses the same procedure as an inexact Newton’s method, except, since the function we use here isb(n) − Ax for n = 1, 2, · · · , we don’t need to compute any Jacobian and, in fact, the Jacobian is thematrix A. For this reason, we also call this iterative method a "Jacobian-free method".

Since the JFHSS scheme uses many HSS inner iterations, one may use another splitting schemeinstead of the HSS method. In fact, if any faster and less time-consuming splitting method is availablethat satisfies the convergence theorem, presented in the next section, then it can be used insteadof the HSS algorithm. One of these methods that is proposed in [14] is GPSS (generalized positivedefinite and skew-Hermitian splitting) algorithm that uses a positive-definite and skew-Hermitiansplitting scheme instead of a Hermitian and skew-Hermitian one. Let H and S be the Hermitian andskew-Hermitian parts of A; then, the GPSS algorithm splits A as A = P1 + P2 where P1 and P2 are,respectively, positive definite and skew-Hermitian matrices. In fact, we have

P1 = D + 2LG , P2 = K+ L∗G − LG + S, (15)

or

P1 = D + 2L∗G , P2 = K+ LG − L∗G + S, (16)

where G and K are, respectively, Hermitian and Hermitian positive semidefinite matrices of H, that is,H = G +K; in addition, D and LG are the diagonal matrix and the strictly lower triangular matrices ofG, respectively (see [14]).

Thus, to solve the system of linear Equation (5) for an initial guess x0 ∈ Cn, and positiveconstants α and tol, the GPSS iteration scheme (until the stopping criterion is satisfied) computes xl forl = 1, 2, . . . by {

(αI + P1)xl+ 12

= (αI − P2)xl + b,

(αI + P2)xl+1 = (αI − P1)xl+ 12+ b,

(17)

where α is a given positive constant and I denotes the identity matrix. In addition, if, in Algorithm1, we use a GPSS scheme instead of an HSS one, we denote the new method by JFGPSS (Jacobianfree GPSS).

3. Convergence of the New Method

As we mentioned in the first section, for solving a nonlinear system, if one can separate (1) intolinear and nonlinear terms, Ax and φ(x), when Ax is strongly dominant over the nonlinear term,Picard-HSS and nonlinear HSS-like methods can solve the problem. However, in many cases, even forweakly nonlinear ones, they may fail to solve the problems. Thus, to obtain a more useful method forsolving (6), based on some splitting methods, we presented a new iterative method. Now, we provethat Algorithm 1 converges to the solution of a weakly nonlinear problem (6). In the following theorem,we prove the convergence of the JFHSS scheme.

Theorem 1. Let x(0) ∈ Cn and ϕ : D ⊂ Cn → Cn be a G-differentiable function on an open set N0 ⊂ Don which ϕ′(x) is continuous and max ‖A−1 ϕ′(x)‖ = L < 1. Let us suppose that H = 1

2 (A + A∗) andS = 1

2 (A− A∗) are the Hermitian and skew-Hermitian parts of the positive definite matrix A and also thatM is an upper bound for ‖A−1G(x(0))‖, and lkn is the number of HSS inner iterations in which the stoppingcriterion (14) is satisfied,

ln∗ >

⎢⎢⎢⎣ ln( (1−η)(1−ηkn−1 )L − 1)

ln(θ)

⎥⎥⎥⎦ , (18)

349


with ln∗ = lim infkn→ ∞

lkn for n = 1, 2, 3, . . ., η is the tolerance in Newton-like intermediate iterations with

L < (1− η)2 and θ = ‖T‖, where T is the HSS inner iteration matrix that can be written as

T = (αI + S)−1(αI − H)(αI + H)−1(αI − S).

Then, the sequence of iteration {x(k)}∞k=0, which is generated by a JFHSS scheme in Algorithm 1,

is well-defined and converges to x∗, satisfying G(x∗) = 0, and also

‖x(n+1) − x(n)‖ ≤ δMρn, (19)

‖x(n+1) − x(0)‖ ≤ δM1− ρ

, (20)

where δ = lim supn→∞

(1 + θln∗ )

1− ηand ρ = lim sup

n→∞ρn for ρn =

(1 + θln∗ )

1− ηL + ηkn−1 .

Proof. Note that ‖T‖ ≤ maxλi∈λ(H)

∣∣∣∣α− λiα + λi

∣∣∣∣ < 1 (see [8]), where λ(H) is the spectral radius of H and α is

a positive constant in HSS inner iterations of JFHSS scheme. Based on Algorithm 1, we can expressx(n+1) as

x(n+1) = x(n)kn= x(n)kn−1 + (I − Tlkn )G′n(x(n)kn−1)

−1Gn(x(n)kn−1)

= x(n)kn−1 + (I − Tlkn )A−1Gn(x(n)kn−1)

= x(n)kn−2 + (I − Tlkn−1)A−1Gn(x(n)kn−2) + (I − Tlkn )A−1Gn(x(n)kn−1)

= x(n)kn−3 + (I − Tlkn−2)A−1G(n)(x(n)kn−3) + (I − Tlkn−1)A−1Gn(x(n)kn−2)

+(I − Tlkn )A−1Gn(x(n)kn−1)

= x(n)0 + (I − Tl1)A−1Gn(x(n)0 ) + (I − Tl2)A−1Gn(x(n)1 ) + · · ·+(I − Tlkn−2)A−1Gn(x(n)kn−3) + (I − Tlkn−1)A−1Gn(x(n)kn−2)

+(I − Tlkn )A−1Gn(x(n)kn−1) = x(n) + ∑knj=1(I − Tlj)A−1Gn(x(n)j−1).

(21)

In the last equality, we used x(n)0 = x(n). If we set η′ = η

cond(A)in (14) instead of η, where

cond(A) = ‖A‖‖A−1‖, then η′ ≤ 1. Because of (14), we have

‖Gn(x(n)kn)‖ ≤ ‖Gn(x(n)kn

)− Gn(x(n)kn−1) + G′n(x(n)kn−1)(x(n)kn− x(n)kn−1)‖

+‖Gn(x(n)kn−1)− G′n(x(n)kn−1)(x(n)kn− x(n)kn−1)‖

= ‖Gn(x(n)kn−1)− A(x(n)kn− x(n)kn−1)‖ ≤ η′‖Gn(x(n)kn−1)‖,

so

‖A−1Gn(x(n)kn)‖ ≤ ‖A−1‖‖Gn(x(n)kn

)‖ ≤ η′‖A−1‖‖Gn(x(n)kn−1)‖≤ η′‖A−1‖‖A‖‖A−1Gn(x(n)kn−1)‖ = η‖A−1Gn(x(n)kn−1)‖.

Therefore, by mathematical induction, we can obtain

‖A−1Gn(x(n)kn)‖ ≤ ηkn‖A−1Gn(x(n)0 )‖. (22)

Then, from (21), and since ‖I − Tlj‖ < 1 + θlj ≤ 1 + θln∗ for j = 1, 2, . . . , kn, we have

350


‖x(n+1) − x(n)‖ ≤ ∑knj=1 ‖I − Tlj‖‖A−1Gn(x(n)j−1)‖

≤ (‖I − Tl1‖+ η‖I − Tl2‖+ η2‖I − Tl3‖+ · · ·+ηkn−2‖I − Tlkn−1‖+ ηkn−1‖I − Tlkn ‖)‖A−1Gn(x(n)0 )‖= (1 + η + η2 + · · ·+ ηkn−2 + ηkn−1)(1 + θln∗ )‖A−1Gn(x(n)0 )‖=

1− ηkn

1− η(1 + θln∗ )‖A−1Gn(x(n)0 )‖.

(23)

Thus, from the last inequality, since Gn(x) = b(n) − Ax, b(n) = ϕ(x(n)), we have

‖x(n+1) − x(n)‖ ≤ 1− ηkn

1− η(1 + θln∗ )‖A−1(b(n) − Ax(n))‖

=1− ηkn

1− η(1 + θln∗ )(‖A−1(ϕ(x(n))− ϕ(x(n−1)))‖+ ‖A−1(ϕ(x(n−1))− Ax(n))‖).

(24)

Then, by using the multivariable Mean Value Theorem (see [15]), we can write

‖A−1(ϕ(x(n))− ϕ(x(n−1)))‖ ≤ maxx∈S

‖A−1 ϕ′(x)‖‖x(n) − x(n−1)‖ = L‖x(n) − x(n−1)‖,

where S = {x : x = tx(n) + (1− t)x(n−1), 0 ≤ t ≤ 1}. Thus,

‖A−1(ϕ(x(n))− ϕ(x(n−1)))‖ ≤ L‖x(n) − x(n−1)‖. (25)

From the right-hand side of (24), using (22) for n− 1, and (25), we have

‖x(n+1) − x(n)‖ ≤=

1− ηkn

1− η(1 + θln∗ )(L‖x(n) − x(n−1)‖+ ‖A−1Gn−1(xn)‖) (26)

≤ 1− ηkn

1− η(1 + θln∗ )(L‖x(n) − x(n−1)‖+ ηkn−1‖A−1Gn−1(x(n−1)

0 )‖).

If in the last inequality of (26), from (23), we use ‖x(n) − x(n−1)‖ ≤ 1− ηkn−1

1− η(1 +

θln−1∗ )‖A−1Gn(x(n−1)0 )‖, then

‖x(n+1) − x(n)‖ ≤1− ηkn

1− η(1 + θln∗ )(L

1− ηkn−1

1− η(1 + θln−1∗ )‖A−1Gn−1(x(n−1)

0 )‖+ ηkn−1‖A−1Gn−1(x(n−1)0 )‖)

≤ 1− ηkn

1− η(1 + θln∗ )(L

1− ηkn−1

1− η(1 + θln−1∗ ) + ηkn−1)‖A−1Gn−1(x(n−1)

0 )‖.

As 1− ηkn < 1, n = 1, 2, · · · and by the definition of ρ and δ, we have

‖x(n+1) − x(n)‖ ≤ δρ‖A−1Gn−1(x(n−1)0 )‖. (27)

By mathematical induction and since ‖A−1G0(x(0)0 )‖ ≤ M,

‖x(n+1) − x(n)‖ ≤ δρn‖A−1G0(x(0)0 )‖ ≤ δMρn, (28)

which yields (19). By the stopping criterion (18), we must have ρ < 1 and then, using (19), it is easyto deduce

351


‖x(n+1) − x(0)‖ ≤ ‖x(n+1) − x(n)‖+ ‖x(n) − x(n−1)‖+ · · ·+ ‖x(1) − x(0)‖ ≤ δM1− ρ

,

which is the relation (20).

Thus, the sequence {x(n)} is in a ball with center x(0) and radius r =δM

1− ρ. From (28),

sequence {x(n)} also converges to its limit point x∗. From the following iteration,

x(n)1 = x(n)0 + (I − Tl1)A−1Gn(x(n)0 ),

when n −→ ∞, ‖x(n)0 − x∗‖ −→ 0, ‖x(n)1 − x∗‖ −→ 0, l1 −→ ∞. Moreover, as ‖T‖ < 1, then Tl1 → 0and we have

G(x∗) = 0,

which completes the proof.

Note that, in some applications, the stopping criterion (18) may be obtained as negative; this showsthat, for all l∗ � 1, we must have ρ < 1.

In addition, it is easy to deduce from the above theorem that any iterative method that its iterationmatrix satisfies in ‖T‖ < 1 can be used instead of the HSS method. For a JFGPSS case, the proof issimilar, except, in the inner iteration, the iterative matrix is

T = (αI + P2)−1(αI − P1)(αI + P1)

−1(αI − P2).

The following result shows the convergence of a JFGPSS algorithm:

Theorem 2. Let x(0) ∈ Cn and ϕ : D ⊂ Cn → Cn be a G-differentiable function on an open set N0 ⊂ D,on which ϕ′(x) is continuous and max ‖A−1 ϕ′(x)‖ = L < 1. Let us suppose that P1 and P2 are generalizedpositive-definite and skew-Hermitian splitting parts of the positive definite matrix A as (15) and (16) and alsothatM is an upper bound for ‖A−1G(x(0))‖; lkn is the number of GPSS inner iterations in which the stoppingcriterion (14) is satisfied,

ln∗ >

⎢⎢⎢⎣ ln( (1−η)(1−ηkn−1 )L − 1)

ln(θ)

⎥⎥⎥⎦ ,

with ln∗ = lim infkn→ ∞

lkn for n = 1, 2, 3, . . ., η is the tolerance in Newton-like intermediate iterations with

L < (1− η)2 and θ = ‖T‖, where T is the GPSS inner iteration matrix that can be written as

T = (αI + P2)−1(αI − P1)(αI + P1)

−1(αI − P2).

Then, the sequence of iteration {x(k)}∞k=0, generated by JFGPSS scheme in Algorithm 1, is well-defined

and converges to x∗, satisfying G(x∗) = 0, and also

‖x(n+1) − x(n)‖ � δMρn,

‖x(n+1) − x(0)‖ � δM1− ρ

,

where δ = lim supn→∞

(1 + θln∗ )

1− ηand ρ = lim sup

n→∞ρn for ρn =

(1 + θln∗ )

1− ηL + ηkn−1 .

Proof. Let us note that, in this theorem, we also have ‖T‖ < 1 (for more details, see [16]). The rest ofthe proof is similar to Theorem 1.

352


In the next section, we apply our new iterative method on some weakly nonlinear systemsof equations.

4. Application

Now, we use JFHSS and JFGPSS algorithms for solving some nonlinear systems. These examplesshow that JFHSS and JFGPSS methods perform better than nonlinear HSS-like and Picard-HSS methods.

Example 1. Consider the following two-dimensional nonlinear convection-diffusion equation

−(uxx + uyy) + q(ux + uy) = f (x, y), (x, y) ∈ Ω

u(x, y) = h(x, y), (x, y) ∈ ∂Ω

where Ω = (0, 1)× (0, 1), ∂Ω is its boundary and q is a positive constant for measuring the magnitude of theconvection term. We solve this problem for each of the following cases:

Case 1 f (x, y) = eu(x,y), h(x, y) = 0.Case 2 f (x, y) = −eu(x,y) − sin(1 + ux(x, y) + uy(x, y)), h(x, y) = −ex+y.

To discretize this convection-diffusion equation, for the convective term, we use a centraldifference method while, for the diffusion term, we use a five-point finite difference method. Theseyield the following nonlinear system

H(u) = Mu + h2ψ(u), (29)

where h =1

N + 1is the equidistance step-size with N as a known natural number and M = AN ⊗

IN + AN ⊗ IN , B = CN × CN with tridiagonal matrices AN = tridiag(−1− qh/2, 2, 1 + qh/2), CN =

tridiag(−1/h, 0, 1/h) and IN is N × N identity matrix. For case 1, we have ψ(u) = −ϕ(u) and, forcase 2, ψ(u) = sin(1 + Bu) + ϕ(u), where ϕ(u) = (eu1 , eu2 , ..., eun)T ; moreover, ⊗ is the Kroneckerproduct symbol, n = N × N and sin(u) means (sin(u1), sin(u2), · · · , sin(un))

T . To apply Picard-HSS,nonlinear HSS-like, JFHSS and JFGPSS methods for solving (29), the stopping criterion for the outeriteration in all methods is chosen as

‖ Mu(n) + h2ψ(u(n)) ‖‖ Mu(0) + h2ψ(u(0)) ‖ ≤ 10−12. (30)

Meanwhile, the Newton-like iteration (in JFHSS and JFGPSS methods) is

‖ Gn(u(n)kn

) ‖‖ Gn(u

(n)0 ) ‖

≤ 10−1, (31)

and also the stopping criterion for HSS and GPSS processes in each Newton-like inner iteration is

‖ Gn(u(n)k )− As(n)k,lkn

‖≤ η‖Gn(u(n)k )‖, (32)

where {u(n)} is the sequence generated by the JFHSS method. kn and lkn are, respectively, the numberof Newton-like inner iterations and HSS and GPSS inner iterations, required for satisfying Relations (31)and (32).

Moreover, to avoid computing the Jacobian in Picard-HSS method, we propose the followingstopping criterion for inner iterations

‖ G(u(n)) + As(n)ln‖≤ η‖G(u(n))‖. (33)

353


In order to use a JFGPSS method, we apply the following decomposition on matrix M in Equation (29),

P1 = D + 2LG , P2 = L∗G − LG + S. (34)

In addition,K = 0, so G = H is the Hermitian part of M and S =12(M−M∗) is the skew Hermitian

part of M.Numerical results for q = 1000, q = 2000 and initial points u(0) = 1, u(0) = 4× 1 for both cases

and u(0) = 12× 1 for case 1 and u(0) = 13× 1 for case 2 and different values of N for JFHSS, JFGPSS,nonlinear HSS-like and Picard-HSS schemes are reported in Tables 1 and 2. Other numerical resultssuch as CPU-time (total CPU time), the number of outer and inner iteration steps (denoted as ITout

and ITinn, respectively), and the norm-2 of the function at the last step (denoted by ‖F(u(n))‖) are alsopresented in these tables. For JFHSS and JFGPSS algorithms, the values of ITint and ITinn are reported.The former is the obtained number when total inner HSS or GPSS iteration is used in Newton-likeiterations, divided by the sum of total Newton-like iterations, while the latter is the total number ofintermediate iterations of the Newton-like method.

Except for u(0) = 1, which is relatively close to the solution (in case 1, the real solution u isnear zero and, in case 2, almost for all coordinates of the solution, ui, i = 1, 2, · · · , n, 0 ≤ ui ≤ 1),the nonlinear HSS-like method for other initial points of Tables 1 and 2 could not perform the iterationsat all, but JFHSS and JFGPSS methods for all points in both cases could easily solve the problem.Picard-HSS for these three initial points could not solve the problem and, in all cases, fails to solve theproblem, especially for q > 500.

Numerical results show that the inner iterations for both JFHSS and nonlinear HSS-like are almostthe same but for JFGPSS is less than these two methods. For example, in Table 1, for u(0) = 1, q = 1000and N = 40, the number of inner iterations for JFHSS and JFGPSS methods are, respectively, 133 and96 and this number for total iterations in the nonlinear HSS-like method (consider that there is only onekind of iteration in a nonlinear HSS-like method) is 127. However, the nonlinear HSS-like method needsto evaluate a greater number of the nonlinear term ψ(u) than the JFHSS method (for the JFHSS method,only 12 function evaluations are required compared to 254 function evaluations for the nonlinearHSS-like method). Thus, JFHSS and JFGPSS methods can significantly reduce the computationalcost of evaluation of the nonlinear term, especially when the nonlinear part is so complicated, e.g.,in Example 2, the difference between the computational cost of the nonlinear HSS-like method and theJFHSS method has increased, since the problem has a more complicated nonlinear term.

It must be noted that, in the inner iteration, for solving the linear systems related to the Hermitianpart (in HSS scheme) and the skew-Hermitian part (in both HSS and GPSS schemes), we have employedrespectively the conjugate gradient (CG) method and the Lanczos method (for more details, see [17]).

In this example, η = tol was used for all steps; in most cases, we obtained equal Newton-likeand outer iterations at each step; however, in general, choosing equal η and tol does not alwayslead to equal Newton-like and outer iterations. For example, in cases that nonlinearity increases(e.g., when we choose initial value u(0) = 12× 1, in the first steps, the nonlinear term h2ψ(u) is so big)result in a different number of Newton-like and outer iterations. In all tables of this paper, a, b denotethe number a · 10b.

354


Table 1. Results for JFHSS, JFGPSS, nonlinear HSS-like and Picard-HSS methods of Example 1, Case 1(η = tol = 0.1).

N 30 40 60 70 80 100

q = 1000, JFHSS CPU 0.65 1.81 7.46 13.21 24.45 59.32u(0) = 1 ITout 12 12 12 12 12 12

ITint 12 12 12 12 12 12ITinn 9 11.08 10.75 10.75 10.41 10.91

‖F(u(n))‖ 1.86, −11 3.35, −11 1.70, −11 3.41, −11 3.54, −11 2.43, −11JFGPSS CPU 0.63 1.46 5.79 9.84 17.28 44.50

ITout 12 12 11 11 11 11ITint 14 12 11 11 11 11ITinn 8.78 8 7.64 7.45 7.90 8.73

‖F(u(n))‖ 5.45, −11 1.89, −11 7.69, −11 1.02, −10 9.63, −11 5.09, −11Nonlinear HSS-like CPU 0.82 2.03 8.26 14.60 24.65 61.35

IT 129 127 123 124 128 126‖F(u(n))‖ 1.45, −10 1.53, −10 1.25, −10 1.10, −10 8.60, −11 8.91, −11

Picard-HSS - - - - - - -


ITint 12 12 12 12 12 12ITinn 16.08 14.67 14.25 14.17 14 14.08

‖F(u(n))‖ 1.47, −10 9.30, −11 7.92, −11 8.80, −11 9.56, −11 6.56, −11JFGPSS CPU 0.85 2.20 8.57 14.26 23.50 54.90

ITout 12 12 12 12 12 12ITint 12 12 12 12 12 12ITinn 14.42 12.42 10.58 10 9.84 9.91


IT 188 172 167 166 165 165‖F(u(n))‖ 3.24, −10 2.50, −10 2.07, −10 2.32, −10 2.037, −10 1.81, −10

Picard-HSS - - - - - - -

q = 1000, JFHSS CPU 0.80 2.24 9.34 14.56 23.77 60.21u(0) = 4× 1 ITout 12 12 12 12 12 12

ITint 12 12 12 12 12 12ITinn 11.08 11 10.67 10.75 10.50 11.25

‖F(u(n))‖ 1.94, −10 1.70, −10 9.33, −11 9.94, −11 1.15, −10 8.77, −11JFGPSS CPU 0.56 1.51 6.55 12.47 21.03 55.50

ITout 12 12 11 12 11 11ITint 12 12 11 12 11 11ITinn 8.92 8.34 8.72 8.75 9.55 10.63

‖F(u(n))‖ 9.76, −11 7.68, −11 4.60, −10 6.35, −11 3.73, −10 3.78, −10Nonlinear HSS-like - - - - - - -

Picard-HSS - - - - - - -


ITint 12 12 12 12 12 12ITinn 16.08 14.67 14.25 14.17 14 14.08

‖F(u(n))‖ 5.88, −10 3.71, −10 3.20, −10 3.57, −10 3.75, −10 2.69, −10JFGPSS CPU 0.85 2.20 8.58 14.02 23.22 54.94



Picard-HSS - - - - - - -


ITint 14 14 14 14 14 14ITinn 10.85 12.83 11.28 11.71 11.64 12.93

‖F(u(n))‖ 1.47, −8 1.05, −8 7.50, −9 4.55, −9 3.08, −9 3.29, −9JFGPSS CPU 0.66 1.70 7.95 14.30 25.44 63.80



Picard-HSS - - - - - - -

355


Table 1. Cont.

N 30 40 60 70 80 100


ITint 14 14 14 13 14 13ITinn 14.93 14.36 14.86 14.62 14.57 15

‖F(u(n))‖ 2.06, −8 1.48, −8 9.76, −9 6.96, −9 6.58, −9 5.72, −9JFGPSS CPU 0.95 2.45 10.03 17.81 29.06 69.57



Picard-HSS - - - - - - -

Table 2. Results for JFHSS, JFGPSS, nonlinear HSS-like and Picard-HSS methods of Example 1, Case 2(η = tol = 0.1).

N 30 40 60 70 80 100


ITint 12 12 12 13 13 13ITinn 11.25 11.41 11.92 11.23 10.92 12

‖F(u(n))‖ 1.64, −10 1.18, −10 1.43, −11 1.32, −11 1.95, −11 1.56, −11JFGPSS CPU 0.57 1.53 7.19 12.59 19.42 53.59

ITout 11 11 11 11 11 12ITint 12 12 12 12 12 13ITinn 9 8.25 8.75 8.92 8.25 9


IT 128 128 123 124 121 126‖F(u(n))‖ 1.81, −10 1.43, −10 1.25, −10 1.10, −10 1.15, −10 1.06, −10

Picard-HSS - - - - - - -


ITint 12 12 12 12 13 13ITinn 16 15 14.50 14.67 14.30 14.62

‖F(u(n))‖ 2.49, −10 2.26, −10 2.03, −10 2.30, −10 2.61, −11 1.74, −11JFGPSS CPU 0.88 2.26 8.61 15.08 25.83 60.7

ITout 11 11 11 11 12 11ITint 12 12 12 12 13 12ITinn 14.33 12.08 10.58 10.66 9.69 11


IT 187 171 166 166 164 164‖F(u(n))‖ 3.68,-10 3.07, −10 2.48, −10 2.17, −10 2.42, −10 2.13, −10

Picard-HSS - - - - - - -


ITint 12 12 12 12 12 12ITinn 11.41 11.33 11.41 11.75 12.08 12.34

‖F(u(n))‖ 1.62, −10 2.01, −10 1.64, −10 1.92, 10 2.89, −10 2.47, −10JFGPSS CPU 0.69 1.97 8.85 16.53 26.53 70.80

ITout 11 11 11 11 12 11ITint 12 12 12 12 13 12ITinn 10.91 11.16 11 11.42 11.42 12.34


Picard-HSS - - - - - - -

356


Table 2. Cont.

N 30 40 60 70 80 100


ITint 12 12 12 12 12 12ITinn 15.92 15.08 14.50 14.58 14.66 14.92

‖F(u(n))‖ 9.65, −10 4.15, −10 4.31, −10 4.40, −10 3.82, −10 3.39, −10JFGPSS CPU 0.88 2.18 8.75 14.60 25.05 64.79



Picard-HSS - - - - - - -


ITint 14 14 14 14 14 14ITinn 10.85 12.83 11.28 11.71 11.64 12.92

‖F(u(n))‖ 1.44, −8 1.36, −8 7.47, −9 4.56, −9 3.8, −9 3.54, −9JFGPSS CPU 0.65 1.74 8.00 14.54 25.02 64.19



Picard-HSS - - - - - - -


ITint 14 14 14 14 14 14ITinn 14.93 14.35 14.43 14.62 14.57 15

‖F(u(n))‖ 2.01, −8 1.49, −8 8.73, −9 6.97, −9 6.57, −9 5.72, −9JFGPSS CPU 0.99 2.41 10.15 17.98 29.33 67.45


‖F(u(n))‖ 1.70, −8 1.30,−8 6.02, −9 3.23, −9 6.34, −9 3.21, −9Nonlinear HSS-like - - - - - - -

Picard-HSS - - - - - - -

The optimal value for parameter α that minimizes the boundary of spectral radius of theiteration matrices is important because it also improves the convergence speed of Picard-HSS,nonlinear HSS-like, JFHSS and JFGPSS methods. There are no general results to determine theoptimal α and α, so we need to obtain the optimal values of parameters α and α experimentally.However, Bai and Golub [8] proved that spectral radius of HSS iterative matrix that is obtained from

the coefficient matrix M in (29) is bounded by ‖T‖ ≤ σ(α) ≡ maxλi∈λ(H)

∣∣∣∣α− λiα + λi

∣∣∣∣ < 1, and the minimum

of this bound is obtained whenα = α∗ =

√λmin(H)λmax(H),

where λmin(H) and λmax(H) are, respectively, the smallest and the largest eigenvalues of Hermitianmatrix H. Usually, in an HSS scheme, αopt �= α∗ ≡ argmin

α>0{σ(α)} < 1 and ρ(T(α∗)) � ρ(T(αopt)).

When q or qh/2 is small, σ(α) is close to ρ(T(α)) and in this case α∗ is close to αopt and α∗ can be agood estimation for αopt. However, when q or qh/2 is large (the skew-Hermitian part is dominant),hence σ(α) deviates too much from ρ(T(α)), so using α∗ is not useful. In this case, ρ(T(α)) attains itsminimum at αopt that is far from α∗, but close to qh/2 (see [8]).

In the GPSS case, a spectral radius of T(α) is bounded by ‖V(α)‖, where V(α) = (αI − P1)(αI +P1)

−1. Since ‖V(α)‖2 � 1 (see [18]), GPSS inner iterations unconditionally converge to the exactsolution in each inner iteration of a JFGPSS scheme. However, when P1 ∈ Cn×n is a generalpositive-definite matrix, we do not have any formula to compute α∗ ≡ argmin

α>0{‖V(α)‖} that is

357


the value that minimizes the boundary of iteration matrix T(α), nor do we have a formula for αopt, thevalue that minimizes ‖T(α)‖.

In Table 3, the optimal values of αopt and αopt have been written (tested and optimal α and αopt)that are determined experimentally by using increments as 0.25. In addition, the correspondingspectral radius of the iteration matrices T(α) and T(α) for HSS and GPSS algorithms that are used asinner iterations to solve (29) are reported in this table. One can see that the spectral radius of GPSSmethod in all cases is smaller than HSS scheme, which results in faster convergence.

Table 3. Optimal value of α for HSS and GPSS inner iterations for different values of N and q ofExample 1.

N 30 40 50 60 70 80 90 100

q = 1000 HSS αopt 18 15 10.5 9 8 6 5.75 5.75ρ(T(αopt)) 0.7226 0.6930 0.6743 0.6613 0.6513 0.6485 0.6459 0.6467

α∗ 0.4047 0.3062 0.2462 0.2059 0.1769 0.1551 0.1381 0.1244ρ(T(α∗)) 0.8971 0.9211 0.9360 0.9461 0.9535 0.9590 0.9634 0.9669

qh2

16.1290 12.1951 9.8039 8.1967 7.0423 6.1728 5.4945 4.9505

ρ(T(qh2)) 0.7236 0.6974 0.6783 0.6674 0.6608 0.6574 0.6562 0.6569

GPSS αopt 11.25 9.5 8.5 7.5 7 6.5 6 5.5ρ(T(αopt)) 0.5428 0.5140 0.5076 0.4983 0.4959 0.4902 0.4982 0.4983

q = 2000 HSS αopt 26 22 16 13.5 12 10 8.75 8ρ(T(αopt)) 0.7911 0.7663 0.6499 0.7399 0.0.7373 0.7302 0.7302 0.7242

α∗ 0.1638 0.0938 0.0606 0.0424 0.0313 0.0241 0.0191 0.0155ρ(T(α∗)) 0.9579 0.9757 0.9842 0.9889 0.9918 0.9937 0.9950 0.9959

qh2

32.2581 24.39 19.61 16.3934 14.0845 12.35 10.99 9.9010

ρ(T(qh2)) 0.7953 0.77 0.7512 0.7439 0.7343 0.728 0.7282 0.7270

GPSS αopt 15 13 11 10 9 8 7.5 7ρ(T(αopt)) 0.6424 0.6212 0.6144 0.6063 0.6036 0.6028 0.6090 0.6033

Example 2 ([10]). We consider the two-dimensional nonlinear convection-diffusion equation

−(uxx + uyy) + qex+y(xux + yuy) = ueu + sin(√

1 + u2x + u2

y), (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = (0, 1)× (0, 1), ∂Ω is its boundary and q is a positive constant for measuring magnitude of theconvection term. By applying the upwind finite difference scheme on the equidistance discretization grid (stepsize

h =1

N + 1) with the central difference scheme to the convective term, we obtain a system of nonlinear equations

in the general form (for more details, see [10])

H(x) = Mx− h2ψ(x). (35)

We have selected zero vector u(0) = 0 = (0, 0, · · · , 0)T as the initial guess. In addition, again (31) and (32)are used respectively as the stopping criteria for the inner iterations and Newton-like iterations in the JFHSSmethod and (30) for outer iterations in JFHSS, Picard-HSS and nonlinear HSS-like methods. Moreover, to avoidcomputing Jacobian in Picard-HSS and nonlinear HSS-like methods, we used (33). Similar to Example 1,one can use other iterative methods instead of HSS in Algorithm 1, for which the spectral radius of its iterationmatrix is smaller and thus results in faster convergence.

Numerical results for N = 32, 48, 64, optimal α and different values of q for JFNHSS, Picard-HSSand nonlinear HSS-like schemes are reported in Table 4. In addition, we adopted the experimentally optimalparameters α to obtain the least CPU times for these iterative methods. One can see that JFHSS performs betterthan nonlinear HSS-like and Picard-HSS methods in all cases.

358


Table 4. Results of JFHSS, nonlinear HSS-like and Picard-HSS methods for Example 2 (η = tol = 0.1).

q 50 100 200 400 1200 2000

N = 32 αopt 1.4 1.6 2.5 8 21.5 34JFHSS CPU 1.23 1.42 1.29 1.53 1.71 1.86

ITout 12 12 12 12 12 12ITint 12 12 12 12 12 12ITinn 11.34 11.67 12 12.75 16.25 20.34


IT 129 137 140 146 160 167‖F(u(n))‖ 1.1, −14 2.25, −14 2.31, −14 2.23, −14 2.4, −14 2.3, −14

Picard-HSS CPU 7.96 8.31 7.76 8 8.60 8.86ITout 12 12 12 12 12 12ITinn 121.1 131.91 126.75 145.34 146.34 147

‖F(u(n))‖ 1.1, −14 1.24, −14 1.57, −14 1.96, −14 1.84, −14 1.6, −14

N = 48 αopt 0.8 1.4 2.6 4.8 13 20.5JFHSS CPU 5.25 5.31 5.5 5.93 6.21 6.28



IT 161 209 178 186 201 207‖F(u(n))‖ 1.5, −14 1.59, −14 1.46, −14 1.57, −14 1.615, −14 1.46, −14

Picard-HSS CPU 50.81 50.01 51.85 53.34 56.32 59.95ITout 12 12 12 12 12 12ITinn 177.16 179.1 183.50 189.34 202.75 213.25

‖F(u(n))‖ 7.7, −15 9.67, −15 1.11, −14 1.23, −14 1.22, −14 1.26, −14

N = 64 αopt 0.7 1 1.8 3.3 8.9 14.2JFHSS CPU 21.68 18.23 18.65 19.156 20.53 21.39

ITout 12 12 12 12 12 12ITint 12 12 12 12 12 12ITinn 21 17.39 18.17 18.75 19.91 20.84


IT 246 206 213 221 235 242‖F(u(n))‖ 1.17, −14 1.26, −14 1.26, −14 1.16, −14 1.19, −14 1.22, −14

Picard-HSS CPU 219.54 217.45 266.83 225.37 228.60 248.35ITout 12 12 12 12 12 12ITinn 219.54 248.58 230.75 252 258.75 264.50

‖F(u(n))‖ 6.12, −15 7.7, −15 8.9, −15 1.0, −14 1.1, −14 1.1, −14

5. Conclusions

In this paper, an iterative method based on two-stage splitting methods has been proposed tosolve weakly nonlinear systems and a convergence property of this method has been investigated.This method is a combination of an inexact Newton method, Hermitian and skew-Hermitian splitting(or generalized positive definite and skew-Hermitian splitting) scheme. The advantage of our newmethod, Picard-HSS and nonlinear HSS-like over the methods like Newton method is that they don’tneed explicit construction and accurate computation of the Jacobian matrix. Hence, computationworks and computer memory may be saved in actual application; however, numerical results showthat JFHSS and JFGPSS methods perform better than the two other ones.

Numerical results show that JFHSS and JFGPSS iteration algorithms are effective, robust,and feasible nonlinear solvers for a class of weakly nonlinear systems. Moreover, employing thesealgorithms to solve nonlinear systems is found to be simple, accurate, fast, flexible, convenient andhave small computation cost. In addition, it must be noted that, even though our inner iterationscheme in this paper are HSS and GPSS methods, another inner iteration solver can be used subject tothe condition that the iteration matrix satisfies in ‖T‖ < 1.

359


Author Contributions: The contributions of authors are roughly equal.


Acknowledgments: The third and fourth authors have been partially supported by the Spanish Ministerio deCiencia, Innovación y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.


References

1. Shen, W.; Li, C. Kantorovich-type convergence criterion for inexact Newton methods. Appl. Numer. Math.2009, 59, 1599–1611. [CrossRef]

2. An, H.-B.; Bai, Z.-Z. A globally convergent Newton-GMRES method for large sparse systems of nonlinearequations. Appl. Numer. Math. 2007, 57, 235-252. [CrossRef]

3. Eisenstat, S.C.; Walker, H.F. Globally convergent inexact Newton methods. SIAM J. Optim. 1994, 4, 393–422.[CrossRef]

4. Gomes-Ruggiero, M.A.; Lopes, V.L.R.; Toledo-Benavides, J.V. A globally convergent inexact Newton methodwith a new choice for the forcing term. Ann. Oper. Res. 2008, 157, 193–205. [CrossRef]

5. Bai, Z.-Z. A class of two-stage iterative methods for systems of weakly nonlinear equations. Numer. Algorithms1997, 14, 295–319. [CrossRef]

6. Zhu, M.-Z. Modified iteration methods based on the Asymmetric HSS for weakly nonlinear systems.J. Comput. Anal. Appl. 2013, 15, 188–195.

7. Axelsson, O.; Bai, Z.-Z.; Qiu, S.-X. A class of nested iteration schemes for linear systems with a coefficientmatrix with a dominant positive definite symmetric part. Numer. Algorithms 2004, 35, 351–372. [CrossRef]

8. Bai, Z.-Z.; Golub, G.H.; Ng, M.K. Hermitian and skew-Hermitian splitting methods for non-Hermitianpositive definite linear systems. SIAM J. Matrix Anal. Appl. 2003, 24, 603–626. [CrossRef]

9. Li, L.; Huang, T.-Z.; Liu, X.-P. Asymmetric Hermitian and skew-Hermitian splitting methods for positivedefinite linear systems. Comput. Math. Appl. 2007, 54, 147–159. [CrossRef]

10. Bai, Z.-Z.; Yang, X. On HSS-based iteration methods for weakly nonlinear systems. Appl. Numer. Math. 2009,59, 2923–2936. [CrossRef]

11. Bai, Z.-Z.; Migallón, V.; Penadés, J.; Szyld, D.B. Block and asynchronous two-stage methods for mildlynonlinear systems. Numer. Math. 1999, 82, 1–20. [CrossRef]

12. Zhu, M.-Z.; Zhang, G.-F. On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations.J. Comput. Appl. Math. 2011, 235, 5095–5104. [CrossRef]

13. Bai, Z.-Z.; Guo, X.-P. On Newton-HSS methods for systems of nonlinear equations with positive-definiteJacobian matrices. J. Comput. Math. 2010, 28, 235–260.

14. Li, X.; Wu, Y.-J. Accelerated Newton-GPSS methods for systems of nonlinear equations. J. Comput. Anal. Appl.2014 , 17, 245–254.

15. Edwards, C.H. Advanced Calculus of Several Variables; Academic Press: New York, NY, USA, 1973.16. Cao, Y.; Tan, W.-W.; Jiang, M.-Q. A generalization of the positive-definite and skew-Hermitian splitting

iteration. Numer. Algebra Control Optim. 2012, 2, 811–821.17. Bai, Z.-Z.; Golub, G.H.; Ng, M.K. On inexact Hermitian and skew-Hermitian splitting methods for

non-Hermitian positive definite linear systems. Linear Algebra Appl. 2008, 428, 413–440. [CrossRef]18. Bai, Z.-Z.; Golub, G.H.; Lu, L.-Z.; Yin, J.-F. Block triangular and skew-Hermitian splitting methods for

positive-definite linear systems. SIAM J. Sci. Comput. 2005, 26, 844–863.


360

mathematics

Article

Higher-Order Iteration Schemes for Solving NonlinearSystems of Equations

Hessah Faihan Alqahtani 1, Ramandeep Behl 2,* and Munish Kansal 3

1 Department of Mathematics, King Abdulaziz University, Campus Rabigh 21911, Saudi Arabia;[email protected]

2 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia3 School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, India;


Received: 23 August 2019; Accepted: 24 September 2019; Published: 10 October 2019

Abstract: We present a three-step family of iterative methods to solve systems of nonlinear equations.This family is a generalization of the well-known fourth-order King’s family to the multidimensionalcase. The convergence analysis of the methods is provided under mild conditions. The analyticaldiscussion of the work is upheld by performing numerical experiments on some application orientedproblems. Finally, numerical results demonstrate the validity and reliability of the suggested methods.

Keywords: systems of nonlinear equations; King’s family; order of convergence; multipointiterative methods

1. Introduction

System of nonlinear equations (SNEs) finds applications to numerous phenomena in many areasof science and engineering. Given a nonlinear system, F(X) = 0, where F is a nonlinear map fromRk → Rk, we are interested to compute a vector X∗ = (x∗1, x∗2, · · · , x∗k )

T such that F(X∗) = 0 , whereF(X) = ( f1(X), f2(X), . . . , fk(X))T is a Fréchet differentiable function and X = (x1, x2, . . . , xk)

T ∈ Rk.The classical Newton’s method [1] is the most famous procedure to solve SNEs. It is given by

X(k+1) = X(k) − {F′(X(k))}−1F(X(k)), k = 0, 1, 2, . . . . (1)

It converges quadratically if the function F is continuously differentiable and the initialapproximation is close enough. In the literature, there are variety of higher-order methods that improvethe convergence order of Newton’s scheme. For example, several authors have proposed cubicallyconvergent methods [2–5] requiring computation of 2-F′ (2-F′ stand for F′ two times), 1-F (1-F stands forF one time), and two matrix inversions per step. In [6], the authors developed another family of methodsof order three, one of which requires one 1-F and 3-F′, whereas the other requires 1-F and 4-F′ evaluationsand two matrix inversions per iteration. In [7], Darvishi and Barati utilized 2-F, 2-F′ and two matrixinversions per step to propose a new third-order scheme. Similarly, several third-order methods havebeen proposed in [8,9] that require 2-F, 1-F′, and one matrix inversion. Babajee et al. [10] presented amethod having convergence order four which consumes 1-F, 2-F′ and two matrix inversions per iteration.Another fourth-order method is developed in [11] using two evaluations of the function and the Jacobianand one matrix inversion, whereas the authors of [12] propose another fourth-order method, utilizing 3-F,1-F′, and one matrix inversion per iteration. Another fifth-order method in [13] requires three evaluationsof the function and only one Jacobian evaluation, with the solution of three linear systems with the samematrix of coefficients per iteration.

Mathematics 2019, 7, 937; doi:10.3390/math7100937 www.mdpi.com/journal/mathematics

361


In pursuit of faster algorithms, researchers have also developed fifth and sixth-order methods, forexample, in [6,14–16]. In [15], Narang et al. extended the existing Babajee’s fourth-order scheme [17] tosolve SNEs and developed a sixth-order convergent family of Chebyshev-Halley type methods. Theirscheme requires two F , two F′ evaluations, and the solution of two linear systems per iteration. One cannotice that although the researchers are making an attempt to improve the order of convergence of aniterative method, it mostly leads to increase in the computational cost per iteration. The computationalcost is especially high if the method involves the use of second order Fréchet derivative F′′(X). This is amajor limitation of the higher-order methods. Thus, although developing new iterative methods, weshould try to keep the computational cost low. With this intention, we have made an attempt to developa family of three-step sixth-order family of methods requiring two F, two F′ and one matrix inversion periteration. This family of methods are compared to be more efficient than existing methods. These havebeen found to be effective in solving particularly large-scale nonlinear systems.

The outline of the manuscript is as follows. In Section 2, a new class of new sixth-order scheme andits convergence analysis is presented. In Section 3, we present numerous illustrative examples to validatethe theoretical results. Finally, Section 4 contains some conclusions.

2. Design of the King’s Family for Multidimensional Case

In this section, we proposed a new three-point extension of King’s method [18–21] having sixth-orderconvergence. For this purpose, we consider the well-known fourth-order King’s method, which isgiven by

yk = xk − f (xk)

f ′(xk),

xk+1 = yk − 1 + αuk1 + (α− 2)uk

f (yk)

f ′(xk),

(2)

where α is a real parameter and uk = f (yk)f (xk)

. For α = 0, one can obtain the well-knownOstrowski’s method [22–24].

Let us now modify the method (2) for SNEs by rewriting the scheme as follows,

uk =f (yk)− f (xk) + f (xk)

f (xk)

=f (yk)− f (xk)

f (xk)+ 1

= 1− f (yk)− f (xk)

(yk − xk) f ′(xk)

= 1− f ′(xk)−1[yk, xk; f ],

where [yk, xk; f ] = f (yk)− f (xk)yk−xk

. Finally, we can rewrite the above scheme (2) for SNEs with one additionalsub-step in the following manner,

y(k) = x(k) − F′(x(k))−1F(x(k)),

z(k) = y(k) − (I + (α− 2)U(k))−1(I + αU(k))F′(x(k))−1F(y(k)),

x(k+1) = z(k) −([y(k), z(k); F]

)−1F(z(k)),

(3)

where [·, ·; F] is a finite difference of order one and α is a free disposable parameter with U(k) = I −[x(k), y(k); F]F′(x(k))−1. In addition, F[Yn, Xn] is a finite difference of order one.

362


Now, it is necessary to analyze the convergence conditions of this modified King’s class of methods.In Theorem 1, we demonstrate the convergence order of the above scheme (3). We have used the followingprocedures [25] to prove the convergence results.

Let F : Ω ⊆ Rk −→ Rk be sufficiently differentiable in Ω. Now, we define the qth derivative ofF at ω ∈ Ω, q ≥ 1. It can be viewed as a q-linear function F(q)(ω) : Rk × · · · × Rk −→ Rk, such thatF(q)(ω)(v1, . . . , vq) ∈ Rk. It is easy to observe that

1. F(q)(ω)(v1, . . . , vq−1, ·) ∈ L(Rk).2. F(q)(ω)(vσ(1), . . . , vσ(q)) = F(q)(ω)(v1, . . . , vq), for all permutation σ of {1, 2, . . . , q}.

Using the above relations, we can introduce the following notation,

(a) F(q)(ω)(v1, . . . , vq) = F(q)(ω)v1 . . . vq.(b) F(q)(ω)vq−1F(p)vp = F(q)(ω)F(p)(ω)vq+p−1.

Now, applying Taylor’s expansion for ξ∗ + h ∈ Rk in the neighborhood of a solution ξ∗ of the givenlinear system, one can get

F(ξ∗ + h) = F′(ξ∗)[

h +p−1

∑q=2

Cqhq

]+ O(hp), (4)

where Cq = (1/q!)[F′(ξ∗)]−1F(q)(ξ∗), q ≥ 2. We note that Cqhq ∈ Rk as F(q)(ξ∗) ∈ L(Rk × · · · ×Rk,Rk),and [F′(x)]−1 ∈ L(Rk).

Similarly, we can express F′ as

F′(ξ∗ + h) = F′(ξ∗)[

I +p−1

∑q=2

qCqhq−1

]+ O(hp−1), (5)

where I denotes the identity matrix. Therefore, qCqhq−1 ∈ L(Rk). From Equation (5), we obtain

[F′(ξ∗ + h)]−1 =[

I + X2h + X3h2 + X4h4 + · · ·][F′(ξ∗)]−1 + O(hp), (6)

whereX2 = −2C2,X3 = 4C2

2 − 3C3,X4 = −8C3

2 + 6C2C3 + 6C3C2 − 4C4,...

Let us denote e(k) = x(k) − ξ∗ as the error at the kth iteration. Then, the error equation is given as follows,

e(k+1) = M(e(k))p + O((e(k))p+1),

where, M is a p-linear function M ∈ L(Rk × · · · ×Rk,Rk). Here, p is the order of convergence and (e(k))p

is a column vector (

p︷︸︸︷e(k), e(k), · · · , e(k))T .

Theorem 1. Let F : Ω ⊆ Rk → Rk be a sufficiently differentiable function defined on a convex set Ω containingthe zero ξ∗. Let us assume that F′(x) is continuous and non-vanishing at ξ∗. If the initial guess x(0) is closeenough to ξ∗, the iterative scheme (3) attains sixth-order convergence for each α.

363


Proof. Let e(k) = x(k) − ξ∗ be the error at the kth-iteration. Now, expanding F(x(k)) and F′(x(k)) usingTaylor’s expansion in a neighborhood of ξ∗, we get

F(x(k)) = F′(ξ∗)[e(k) + C2(e(k))2 + C3(e(k))3 + C4(e(k))4 + C5(e(k))5 + C6(e(k))6

]+ O((e(k))7) (7)

and

F′(x(k)) = F′(ξ∗)[

I + 2C2e(k) + 3C3(e(k))2 + 4C4(e(k))3 + 5C5(e(k))4 + 6C6(e(k))5]+ O((e(k))6), (8)

where I is the identity matrix of size n× n and Ck =1k! F′(ξ∗)−1F(k)(ξ∗), k ≥ 2.

With the help of above expression (8), we have

F′(x(k))−1 =[

I − 2C2e(k) + Δ0(e(k))2 + Δ1(e(k))3 + Δ2(e(k))4 + Δ3(e(k))5 + Δ4(e(k))6 + O((e(k))7]

F′(ξ∗)−1 (9)

where Δi = Δi(C2, C3, . . . , C6), for example, Δ0 = 4C22 − 3C3, Δ1 = −(8C3

2 − 6C2C3 − 6C3C2 +

4C4), Δ2 = 8C2C4 + 9C23 + 8C4C2 − 12C2

2C3 − 12C2C3C2 − 12C3C22 + 16C4

2 − 5C5, Δ3 = 10C2C5 +

12C3C4 + 12C4C3 + 10C5C2− 16C22C4− 18C2C2

3 − 16C2C4C2− 18C3C2C3− 18C23C2− 16C4C2

2 + 24C32C3 +

24C22C3C2 + 24C2C3C2

2 + 24C3C32 − 32C5

2 − 6C6, etc.From expressions (7) and (9), we yield

F′(x(k))−1F(x(k)) =e(k) + Θ0(e(k))2 + Θ1(e(k))3 + Θ2(e(k))4 + Θ3(e(k))5 + Θ4(e(k))6 + O((e(k))7). (10)

where Θj = Θj(C2, C3, . . . , C6), for example, Θ0 = −C2, Θ1 = 2C22 − 2C3, Θ2 = −(4C3

2 − 4C2C3 −3C3C2 + 3C4), Θ3 = 6C2C4 + 6C2

3 + 4C4C2 − 8C22C3 − 6C2C3C2 − 6C3C2

2 + 8C42 − 4C5, Θ4 = 8C2C5 +

9C3C4 + 8C4C3 + 5C5C2 − 12C22C4 − 12C2C2

3 − 8C2C4C2 − 12C3C2C3 − 9C23C2 − 8C4C2

2 + 16C32C3 +

12C22C3C2 + 12C2C3C2

2 + 12C3C32 − 16C5

2 − 5C6, etc.By inserting the expression (10) in the first substep of (3), we obtain

y(k) − ξ∗ = −Θ0(e(k))2 −Θ1(e(k))3 −Θ2(e(k))4 −Θ3(e(k))5 −Θ4(e(k))6 + O((e(k))7). (11)

which further produces

F(y(k)) =F′(ξ∗)[−Θ0(e(k))2 −Θ1(e(k))3 + (C2Θ2

0 −Θ2)(e(k))4 + (2C2Θ0Θ1 −Θ3)(e(k))5+

−(

C3Θ30 − C2(Θ2

1 + 2Θ0Θ2) + Θ4

)(e(k))6 + O((e(k))7)

] (12)

and

U(k) = I − F′(x(k))−1[x(k), y(k); F] = C2e(k) + (C2Θ0 − 2C22 + 2C3)(e(k))2 +

(− 2C2

2Θ0 + C2(Θ1 − 7C3) + C3Θ0 + 4C32

+ 3C4

)(e(k))3 +

(4C3

2Θ0 + C22(20C3 − 2Θ1) + C2(−5C3Θ0 − 10C4 + Θ2) + C4Θ0

+ C3(Θ1 −Θ20)− 8C4

2 − 6C23 + 4C5

)(e(k))4 +

[− 8C4

2Θ0 + C32(4Θ1 − 52C3)

+ 2C22(8C3Θ0 + 14C4 −Θ2) + C2

(C3(2Θ2

0 − 5Θ1)− 6C4Θ0 + 33C23 − 13C5 + Θ3

)− C4Θ2

0 − 3C23Θ0 + C5Θ0 + C4Θ1 + C3(−17C4 − 2Θ0Θ1 + Θ2) + 16C5

2 + 5C6

](e(k))5

+ O((e(k))6).

(13)

By using expressions (9), (12), and (13), we obtain(I + (α− 2)U(k)

)−1(I + αU(k)

)F′(x(k))−1F(y(k)) = −Θ0(e(k))2 −Θ1(e(k))3 + (2αC2

2Θ0 − C2Θ20

− C3Θ0 −Θ2)(e(k))4 + O((e(k))5) (14)

364


By adopting expressions (11)–(14) in the scheme (3), we have

z(k) − ξ∗ = τ1(e(k))4 + τ2(e(k))5 + τ3(e(k))6 + O((e(k))7). (15)

where τj = τj(Θ0, Θ1, . . . , Θ4, C2, C3, . . . , C6, α), j = 1, 2, 3, for example, τ1 = Θ0(−2αC22 + C2Θ0 + C3).

Now, expanding the F(z(k)) in a neighborhood of ξ∗, we have

F(z(k)) = F′(ξ∗)[τ1(e(k))4 + τ2(e(k))5 + τ3(e(k))6 + O((e(k))7)

], (16)

which further produces with the help of expression (12)[z(k), y(k); F

]−1F(z(k)) = τ1(e(k))4 + τ2(e(k))5 + (C2Θ0τ1 + τ3) (e(k))6 + O((e(k))7). (17)

By using equation (17) in the final substep of (3), we have

e(j+1) =(

τ1(e(k))4 + τ2(e(k))5 + τ3(e(k))6)−(

τ1(e(k))4 + τ2(e(k))5 + (C2Θ0τ1 + τ3)(e(k))6)+ O((e(k))7)

= −(C2Θ0τ1)(e(k))6 + O((e(k))7)

= C32((2α + 1)c2

2 − c3)(e(k))6 + O((e(k))7).

(18)

Therefore, the scheme (3) has sixth-order convergence.


Here, we checked the efficiency and effectiveness of our scheme on real-life and standard academictest problems. Therefore, we consider five number of the examples’ details, as seen in the examples (1)–(5).Further, we also depicted the starting points and zeros of the considered nonlinear system in theexamples (1)–(5). Now, we employ our sixth-order scheme (3), called (PM), to verify the computationalperformance of them with existing methods considered in the previous section.

Now, we compare (3) with a sixth-order family proposed by Abbasbandy et al. [26] andHueso et al. [27]. We choose their best expressions (8) and (14–15)

(for t1 = − 9

4 and s2 = 98), respectively

denoted by (AM6) and (HM6). Moreover, a comparison of a newly proposed scheme has been donewith the sixth-order family of iterative method proposed by Sharma and Arora [28] and Wang and Li [29],out these works we choose two methods, namely, (13) and (6), respectively, called (SM6) and (WM6).Finally, we compare (3) with sixth-order methods suggested by Mona et al. [15] and Lotfi et al. [16], weconsider methods (3.1)

(for λ = 1, β = 2, p = 1 and q = 3

2)

and (5), respectively, called by (MM6) and(LM6). All the iterative expressions are mentioned below.

Method AM6:

y(k) = x(k) − 23

F′(x(k))−1F(x(k)),

z(k) = x(k) −[

I +218

F′(x(k))−1F′(y(k))− 92(

F′(x(k))−1F′(y(k)))2

+158(

F′(x(k))−1F′(y(k)))3]

F′(x(k))−1F(x(k)),

x(k+1) = z(k) −[3I − 5

2F′(x(k))−1F′(y(k)) + 1

2(

F′(x(k))−1F′(y(k)))2]

F′(x(k))−1F(z(k)).

(19)

Method HM6:y(k) = x(k) − F′(x(k))−1F(x(k)),

H(x(k), y(k)) = F′(x(k))−1F(y(k)),

Gs(x(k), y(k)) = s1 I + s2H(y(k), x(k)) + s3H(x(k), y(k)) + s4H(y(k), x(k))2,

z(k) = x(k) − Gs(x(k), y(k))F′(x(k))−1F(x(k)),

x(k+1) = z(k).

(20)

365


where s1 s2, s3, and s4 are real numbers.

Method SM6:

y(k) = x(k) − 23

F′(x(k))−1F(x(k)),

z(k) = φ(4)4 (xk, yk),

x(k+1) = z(k) −[sI + tF′(x(k))−1F′(y(k))

]F′(x(k))−1F(z(k)),

(21)

where s and t are real parameters.

Method WM6:y(k) = x(k) − F′(x(k))−1F(x(k)),

z(k) = y(k) −[2I − F′(x(k))−1F′(y(k))

]F′(x(k))−1F(y(k)),

x(k+1) = z(k) −[2I − F′(x(k))−1F′(y(k))

]F′(x(k))−1F(z(k)).

(22)

Method MM6:

y(k) = x(k) − 23

F′(x(k))−1F(x(k)),

z(k) = x(k) −[

I +1

2β

(I − λ

βG(x(k))

)H(G(x(k)))u(x(k))

],

x(k+1) = z(k) −[

pI + qG(x(k))]

F′(x(k))−1F(z(k)),

(23)

where λ, β, p, and q are real numbers.

Method LM6:

y(k) = x(k) − F′(x(k))−1F(x(k)),

z(k) = x(k) − 2(F′(x(k) + F′(y(k))−1F(x(k)),

x(k+1) = z(k) −[7

2I − 4F′(x(k))−1F′(y(k) + 3

2F′(x(k))−1F′(y(k))2

]F′(x(k))−1F(z(k)).

(24)

The abscissas tj and the weights wj are known and depicted in the Table 1 when t = 8(for themore details please have a look at Example 1). In Tables 2–6, we mention the number of iterationindexes (n), residual errors (‖F(x(k))‖), error ‖x(k+1) − x(k)‖ and computational convergence order

ρ∗ ≈ log[‖x(k+1)−x(k)‖/‖x(k)−x(k−1)‖

]log[‖x(k)−x(k−1)‖/‖x(k−1)−x(k−2)‖

] . In addition, the value of η is the last calculated value of ‖x(k+1)−x(k)‖‖x(k)−x(k−1)‖6 .

Finally, the comparison on the basis of number of iterations taken by different methods on numericalExamples 1–5 is also depicted in Table 7.

All the computations have been done with multiple precision arithmetic with 1000 digits of mantissa,which minimize round-off errors in Mathematica-9. Here, a (±b) is a × 10(±b) in all the tables. Thestopping criteria for the programming is defined as follows,

(i) ‖F(x(k))‖ < 10−100 and (ii) ‖x(k+1) − x(k)‖ < 10−100.

Example 1. Let us consider the Hammerstein integral equation (see [1], pp. 19–20) given as follows,

y(s) = 1 +15

∫ 1

0F(s, t)y(t)3dt,

where y ∈ C[0, 1], s, t ∈ [0, 1], and the kernel F is

F(s, t) =

{(1− s)t, t ≤ s,

s(1− t), s ≤ t.

366


Now, using the Gauss Legendre formula, we transform the above equation into a finite-dimensional problem, which

is given as∫ 1

0 f (t)dt �8

∑j=1

wj f (tj), where tj and wj denote abscissas and weights, respectively. Now t′sj and w′sj

are determined for t = 8 by Gauss Legendre quadrature formula. Let us call y(ti) by yi(i = 1, 2, . . . , 8), then weget the following nonlinear system,

5yi − 5−8

∑j=1

aijy3j = 0, where i = 1, 2, . . . , 8,

where aij =

{wjtj(1− ti), j ≤ i,

wjti(1− tj), i < j.

Here, the abscissas tj and the weights wj are known and depicted in the Table 1 when t = 8.The convergence of the methods towards the root

ξ∗ = (1.00209 . . . , 1.00990 . . . , 1.01972 . . . , 1.02643 . . . , 1.02643 . . . , 1.01972 . . . , 1.00990 . . . , 1.00209 . . . )T ,

is tested in the Table 2 on the choice of the initial guess x(0) =(

12 , 1

2 , 12 , 1

2 , 12 , 1

2 , 12 , 1

2

)T. The numerical results

show that the proposed methods PM16 and PM26 have better residual errors in comparison with the existing ones.In addition, smaller asymptotic error constants also belong to our methods PM16 and PM26.

Table 1. t′sj and w′sj of Gauss–Legendre formula for t = 8.

j tj wj

1 0.01985507175123188415821957 . . . 0.05061426814518812957626567 . . .2 0.10166676129318663020422303 . . . 0.11119051722668723527217800 . . .3 0.23723379504183550709113047 . . . 0.15685332293894364366898110 . . .4 0.40828267875217509753026193 . . . 0.18134189168918099148257522 . . .5 0.59171732124782490246973807 . . . 0.18134189168918099148257522 . . .6 0.76276620495816449290886952 . . . 0.15685332293894364366898110 . . .7 0.89833323870681336979577696 . . . 0.11119051722668723527217800 . . .8 0.98014492824876811584178043 . . . 0.05061426814518812957626567 . . .

Table 2. Comparison of methods on Hammerstein integral equation in Example 1.

Methods k ‖F(x(k))‖ ‖x(k+1) − x(k)‖ ρ∗ ‖x(k+1)−x(k)‖‖x(k)−x(k−1)‖6

AM6

1 1.1(−5) 2.4(−6)2 5.4(−39) 1.2(−39) 6.596956919(−6)3 8.0(−239) 1.7(−239) 5.9991 7.072478176(−6)

HM6

1 3.0(−5) 6.5(−6)2 6.9(−31) 1.5(−31) 1.9947995983 4.4(−159) 9.4(−160) 4.9991 9.220736175(+25)

SM6

1 9.4(−6) 2.0(−6)2 1.5(−39) 3.2(−40) 4.964844066(−6)3 2.7(−242) 5.7(−243) 5.9991 5.324312398(−6)

367


Table 2. Cont.


WM6

1 1.6(−6) 3.3(−7)2 9.0(−45) 1.9(−45) 1.377597868(−6)3 3.3(−274) 7.1(−275) 5.9996 1.431074607(−6)

MM6

1 3.9(−6) 8.2(−7)2 5.8(−36) 1.2(−36) 3.9435081423 4.6(−185) 9.9(−186) 4.9992 2.773373608(+30)

LM6

1 8.6(−6) 1.8(−6)2 8.4(−37) 1.7(−37) 4.445532921(−3)3 1.4(−189) 3.0(−190) 4.9224 1.232404905(+31)

PM16

1 1.7(−7) 1.5(−7)2 1.9(−47) 4.1(−48) 3.291248720(−7)3 7.6(−291) 1.6(−291) 5.9998 3.358306951(−7)

PM26

1 6.6(−7) 1.4(−7)2 1.1(−47) 2.3(−48) 2.820724919(−7)3 2.0(−292) 4.3(−293) 5.9998 2.880288646(−7)

Example 2. Let us consider the Van der Pol equation [30], which governs the flow of current in a vacuum tube,defined as follows,

y′′ − μ(y2 − 1)y′ + y = 0, μ > 0. (25)

Here, boundary conditions are given by y(0) = 0, y(2) = 1. Further, we divide the given interval [0, 2] as follows,

x0 = 0 < x1 < x2 < x3 < · · · < xn, where xi = x0 + ih, h =2n

.

Moreover, we assume thatyi = y(xi), i = 0, 1, . . . , n.

Now, if we discretize the above problem (25) by using the finite-difference formula for the first and second derivatives,which are given by

y′k =yk+1 − yk−1

2h, y′′k =

yk−1 − 2yk + yk+1

h2 , k = 1, 2, . . . , n− 1,

then, we obtain a SNEs of order (n− 1)× (n− 1).

2h2yk − hμ(

y2k − 1

)(yk+1 − yk−1) + 2 (yk−1 + yk+1 − 2yk) = 0, k = 1, 2, . . . , n− 1.

Let us consider μ = 12 and initial approximation y(0)k = log

(1k2

), k = 1, 2, . . . , n − 1. In this example,

we solve 9× 9 SNEs by taking n = 10. The solution of this problem is

ξ∗ =(− 0.4795 . . . ,−0.9050 . . . ,−1.287 . . . ,−1.641 . . . ,−1.990 . . . ,−2.366 . . . ,−2.845 . . . ,−3.673 . . . ,−6.867 . . . ,

)T .

The numerical results are displayed in Table 3. It is found that the newly proposed methods perform better in allaspects, whereas the existing methods show larger residual errors.

368


Table 3. Comparisons of methods on Van der Pol equation in Example 2.


AM6

1 9.8(+16) 1.2(+6)2 9.3(+15) 5.3(+5) 2.207185393(−31)3 8.9(+14) 2.4(+5) 1.0000 1.115653208(−29)

HM6

1 2.0(+2) 3.4(+1)2 2.6(+0) 1.4(+3) 8.575063031(−7)3 6.9(+7) 8.5(+2) 0.13787 1.044481693(−16)

SM6

1 4.7(+10) 9.4(+3)2 3.9(+9) 4.1(+3) 6.141757152(−21)3 3.3(+8) 1.8(+3) 0.99993 3.757585542(−19)

WM6

1 3.8(+10) 8.7(+3)2 3.3(+9) 3.9(+3) 8.901990983(−21)3 2.9(+8) 1.7(+3) 0.99992 5.216543771(−19)

MM6

1 1.8(+9) 3.2(+3)2 1.4(+8) 1.4(+3) 1.233926627(−18)3 1.1(+7) 6.0(+2) 0.99947 8.435924608(−17)

LM6

1 7.8(+3) 5.1(+1)2 6.0(+2) 1.9(+1) 1.090110495(−9)3 3.9(+1) 5.0(0) 1.3904 9.718708834(−8)

PM16

1 2.6(0) 7.2(−1)2 1.5(−4) 2.8(−5) 2.099986524(−4)3 6.7(−27) 1.6(−27) 5.0546 3.049589936

PM26

1 2.6(−1) 1.2(−1)2 8.3(−9) 1.5(−9) 4.804752944(−4)3 2.8(−48) 7.4(−49) 4.9654 7.413843150(+4)

Example 3. The 2D Bratu problem [31,32] is defined as

uxx + utt + Ceu = 0, on

Ω : (x, t) ∈ 0 ≤ x ≤ 1, 0 ≤ t ≤ 1,

with boundary conditions u = 0 on Ω.

(26)

Using finite difference discretization, a given nonlinear PDE can be reduced to a SNEs. Let Θi,j = u(xi, tj) be thenumerical solution at the grid points of the mesh. Let M1 and M2 be the number of steps in x and t directions,respectively, and h and k be the respective step sizes. To solve the given PDE, we apply the central difference formulato uxx and utt in the following way,

uxx(xi, tj) =Θi+1,j − 2Θi,j + Θi−1,j

h2 , C = 0.1, t ∈ [0, 1] (27)

By using expression (27) in (26) and after some simplification, we have

Θi,j+1 + Θi,j−1 −Θi,j + Θi+1,j + Θi−1,j + h2C exp(

Θi,j

)i = 1, 2, 3, . . . , M1, j = 1, 2, 3, . . . , M2 (28)

By choosing M1 = M2 = 11, C = 0.1, and h = 111 , we obtain a system of nonlinear equations of size 10× 10,

with the initial guess x0 = 0.1(sin(πh)sin(πk), sin(2πh)sin(2πk), . . . , sin(10πh)sin(10πk))T. Numericalestimations are given in Table 4. Numerical results demonstrate that the new methods have much improved errorestimations and computational order of convergence in comparison to its competitors.

369


Table 4. Comparisons of different methods on 2D Bratu problem in Example 3.


AM6

1 4.4(−15) 2.4(−14)2 6.9(−95) 3.5(−94) 1.428095547(−12)3 7.9(−574) 3.9(−573) 5.9994 1.973434769(−12)

HM6

1 2.1(−13) 1.2(−12)2 2.1(−71) 1.2(−70) 7.368055345(−11)3 1.7(−361) 9.3(−361) 4.9997 3.495510769(+1)

SM6

1 4.4(−15) 2.4(−14)2 7.1(−95) 3.6(−94) 1.433541371(−12)3 9.2(−574) 4.5(−573) 5.9994 1.433541371(−12)

WM6

1 8.1(−2) 5.0(−1)2 5.0(−19) 2.9(−18) 1.754949400(−16)3 1.7(−122) 1.0(−121) 5.9999 1.666475363(−16)

MM6

1 7.5(−15) 4.1(−14)2 1.2(−80) 6.4(−80) 1.375781477(+1)3 1.2(−409) 6.3(−409) 4.9997 9.148606524(+66)

LM6

1 6.4(−16) 2.5(−15)2 2.9(−87) 5.3(−87) 1.447342836(−13)3 9.1(−445) 3.3(−444) 4.9853 2.808772449(+1)

PM16

1 5.7(−18) 7.0(−18)2 4.9(−117) 5.9(−117) 5.229619528(−14)3 1.9(−711) 2.3(−711) 5.9999 5.367043263(−14)

PM26

1 5.7(−18) 6.9(−18)2 4.7(−117) 5.6(−117) 1.698178527(−6)3 1.4(−711) 1.6(−711) 5.9999 1.854013522(−6)

Example 4. Consider another typical nonlinear problem, that is, Fisher’s equation [33] with homogeneousNeumann’s BC’s, the diffusion coefficient H is

ut = Huxx + u(1− u) = 0,

u(x, 0) = 1.5 + 0.5cos(πx), 0 ≤ x ≤ 1,

ux(0, t) = 0, ∀ t ≥ 0,

ux(1, t) = 0, ∀ t ≥ 0.

(29)

Again using finite difference discretization, the equation (29) reduces to a SNEs. Consider Θi,j = u(xi, tj) be itsapproximate solution at the grid points of the mesh. Let M1 and M2 be the number of steps in x and t directions,and h and k be the respective step size. Applying central difference formula to uxx, backward difference for ut(xi, tj),and forward difference for ux(xi, tj), respectively, in the following way,

uxx(xi, tj) =Θi+1,j − 2Θi,j + Θi−1,j

h2 ,

ut(xi, tj) =Θi,j −Θi,j−1

kand

ux(xi, tj) =Θi+1,j −Θi,j

h,

(30)

where h = 1M1

, k = 1M2

, t ∈ [0, 1].

370


By adopting expression (30) in (29), after some simplification, we get

Θ1,j −Θi,j−1

k−Θi,j

(1−Θi,j

)− H

Θi+1,j − 2Θi,j + Θi−1,j

h2 , i = 1, 2, 3, . . . , M1, j = 1, 2, 3, . . . , M2 (31)

For the system of nonlinear equations, we considered M1 = M2 = 5, h = 15 , k = 1

5 and H = 1, which reduces to

nonlinear system of size 5× 5, with the initial guess x0 =(1 + i

25)T , i = 1, 2, . . . , 25. All the numerical results

are shown in Table 5. Numerical results show that our methods have better computational efficiency than thealready existing schemes, in terms of residual errors and difference between consecutive approximations.

Table 5. Comparisons of different methods on Fisher’s equation in Example 4.


AM6

1 6.9(−3) 1.5(−3)2 4.2(−21) 6.5(−22) 7.021836209(−5)3 4.5(−131) 7.0(−132) 5.9941 9.022505886(−5)

HM6

1 6.0(−3) 1.3(−3)2 1.3(−18) 2.0(−19) 5.236602433(−2)3 1.8(−97) 2.8(−98) 4.9940 4.016099735(+14)

SM6

1 4.8(−3) 1.0(−3)2 2.9(−22) 4.5(−23) 4.391559871(−5)3 3.0(−138) 4.7(−139) 5.9945 5.610610203(−5)

WM6

1 4.8(−3) 1.0(−3)2 2.9(−22) 4.5(−23) 4.391559871(−5)3 3.0(−138) 4.7(−139) 5.9945 5.610610203(−5)

MM6

1 2.7(−3) 5.5(−4)2 9.5(−21) 1.5(−21) 5.507288873(−2)3 1.6(−108) 2.5(−109) 4.9964 2.350947999(+16)

LM6

1 4.4(−3) 1.0(−3)2 4.6(−20) 7.4(−21) 5.815894917(−3)3 2.2(−107) 1.2(−108) 5.1204 7.024389352(+12)

PM16

1 5.7(−18) 7.0(−18)2 4.9(−117) 5.9(−117) 5.229619528(−14)3 1.9(−711) 2.3(−711) 5.9999 5.367043263(−14)

PM26

1 5.7(−18) 6.9(−18)2 4.7(−117) 5.6(−117) 5.194329469(−14)3 1.4(−711) 1.6(−711) 5.9999 5.331099808(−14)

Example 5. Let us consider the following nonlinear system,

F(X) =

{x2

j xj+1 − 1 = 0, 1 ≤ j ≤ n− 1,

x2nx1 − 1 = 0.

(32)

To obtain a large SNEs 200 × 200, we choose n = 200 and the initial approximation x(0) =

(1.25, 1.25, 1.25, · · · , 1.25(200times))T for this problem. The required solution of this problem isξ∗ = (1, 1, 1, · · · , 1(200times))T. The obtained results can be observed in Table 6. It can be easily seenthat the proposed scheme performs well; in this case, in terms of error estimates as compared to the available methodsof the same nature.

371


Table 6. Comparisons of different methods on Example 5.


AM6

1 5.5(−2) 1.8(−2)2 8.2(−15) 2.7(−15) 7.599151595(−5)3 9.5(−92) 3.2(−92) 5.9993 7.695242316(−5)

HM6

1 3.0(−2) 1.0(−2)2 2.0(−14) 6.7(−15) 6.625126396(−3)3 2.7(−75) 8.9(−76) 4.9997 9.962407479(+9)

SM6

1 3.9(−2) 1.3(−2)2 6.4(−16) 2.1(−16) 4.636076578(−5)3 1.3(−98) 4.3(−99) 6.0000 4.674761498(−5)

WM1 4.2(−2) 1.4(−2)2 1.2(−15) 3.9(−16) 5.254428593(−5)3 5.7(−97) 1.9(−97) 5.9995 5.303300859(−5)

MM6

1 2.7(−2) 8.9(−3)2 5.3(−15) 1.8(−15) 3.463613165(−3)3 1.6(−78) 5.4(−79) 4.9993 1.781568229(+10)

LM6

1 3.4(−2) 1.1(−2)2 2.3(−16) 7.7(−17) 3.721130070(−5)3 2.3(−101) 7.6(−102) 5.9996 3.746683848(−5)

PM16

1 5.5(−3) 1.8(−3)2 3.4(−22) 1.1(−22) 2.944381059(−6)3 1.8(−137) 5.9(−138) 6.0000 2.946278255(−6)

PM26

1 2.8(−3) 9.5(−4)2 2.5(−24) 8.5(−25) 1.178221976(−6)3 1.3(−150) 4.4(−151) 6.0000 1.178511302(−6)

Table 7. Number of iterations taken by different methods on Examples 1–5.

Methods Ex.1 Ex.2 Ex.3 Ex.4 Ex. 5 Total Average

AM6 3 20 3 3 4 33 6.6HM6 3 D 3 3 4 13 * 3.25 *SM6 3 13 3 3 4 26 5.2WM6 3 13 2 3 4 25 5MM6 3 12 3 3 4 25 5LM6 3 7 3 3 3 19 3.8

PM16 3 4 2 2 3 14 2.8PM26 3 4 2 2 3 14 2.8

* means, the total number of iteration calculated on all examples except Example 2, because HM6 isdivergent in Example 2.

4. Conclusions

In this work, we have developed new family of sixth-order iterative methods for solving SNEs,numerically. To check their effectiveness, the proposed scheme is applied on some large-scale systemsarising from various academic problems. Further, the numerical results show that the proposed methodsperform better than already existing schemes in the scientific literature.

Author Contributions: R.B. and M.K.: Conceptualization; Methodology; Validation; Writing–Original DraftPreparation; Writing–Review & Editing. H.F.A.: Review & Editing.

Funding: Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No.G-1349-665-1440.

372


Acknowledgments: This project was supported by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, Jeddah, under grant No. G-1349-665-1440. The authors, therefore, gratefully acknowledge with thanksDSR for technical and financial support.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Ortega J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press:New York, NY, USA, 1970.

2. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method for functions of several variables. Appl. Math.Comput. 2006, 183, 199–208. [CrossRef]

3. Frontini M.; Sormani, E. Third-order methods from quadrature formulae for solving systems of nonlinearequations. Appl. Math. Comput. 2004, 149, 771–782. [CrossRef]

4. Grau-Sánchez, M.; Grau, À; Noguera, M. On the computational efficiency index and some iterative methods forsolving systems of non-linear equations. Comput. Appl. Math. 2011, 236, 1259–1266.

5. Homeier, H.H.H. A modified Newton method with cubic convergence: The multivariable case. Comput. Appl.Math. 2004, 169, 161–169. [CrossRef]

6. Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math.Comput. 2007, 190, 686–698. [CrossRef]

7. Darvishi, M.T.; Barati, A. Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math.Comput. 2007, 188, 1678–1685. [CrossRef]

8. Darvishi, M.T.; Barati, A. A third-order Newton-type method to solve systems of non-linear equations.Appl. Math. Comput. 2007, 187, 630–635.

9. Potra, F.A.; Pták, V. Nondiscrete Induction and Iterarive Processes; Pitman Publishing: Boston, MA, USA, 1984.10. Babajee, D.K.R.; Madhu, K.; Jayaraman, J. On some improved harmonic mean Newton-like methods for solving

systems of nonlinear equations. Algorithms 2015, 8, 895–909. [CrossRef]11. Cordero, A.; Martínez, E.; Torregrosa, J.R. Iterative methods of order four and five for systems of nonlinear

equations. Comput. Appl. Math. 2009, 231, 541–551. [CrossRef]12. Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. Efficient high-order methods based on golden ratio for

nonlinear systems. Appl. Math. Comput. 2011, 217, 4548–4556. [CrossRef]13. Arroyo, V.; Cordero, A.; Torregrosa, J.R. Approximation of artificial satellites’ preliminary orbits: The efficiency

challenge. Math. Comput. Modell. 2011, 54, 1802–1807. [CrossRef]14. Alzahrani, A.K.H.; Behl, R.; Alshomrani, A. Some higher-order iteration functions for solving nonlinear models,

Appl. Math. Comput. 2018, 334, 80–93. [CrossRef]15. Narang, M.; Bhatia, S.; Kanwar, V. New two parameter Chebyshev-Halley like family of fourth and sixth-order

methods for systems of nonlinear equations. Appl. Math. Comput. 2016, 275, 394–403. [CrossRef]16. Lotfi, T.; Bakhtiari, P.; Cordero, A.; Mahdiani, K.; Torregrosa, J.R. Some new efficient multipoint iterative

methods for solving nonlinear systems of equations. Int. J. Comput. Math. 2015, 92, 1921–1934. [CrossRef]17. Babajee, D.K.R. On a two-parameter Chebyshev-Halley like family of optimal two-point fourth order methods

free from second derivatives. Afrika Matematika 2015, 26, 689–695. [CrossRef]18. Campos, B.; Cordero, A.; Torregrosa, J.R.; Vindel, P. Stability of King’s family of iterative methods with memory.

J. Comput. Appl. Math. 2017, 318, 504–514. [CrossRef]19. Cordero, A.; Maimó, J.G.; Torregrosa, J.R.; Vassileva, M.P.; Vindel, P. Chaos in King’s iterative family. Appl. Math.

Lett. 2013, 26, 842–848. [CrossRef]20. Chicharro, F.; Cordero, A.; Torregrosa, J.R. Dynamics of iterative families with memory based on weight

functions procedure. J. Comput. Appl. Math. 2019, 354, 286–298. [CrossRef]21. King, R.F. A family of fourth-order methods for nonlinear equations. SIAM J. Numer. Anal. 1973, 10, 876–879.

[CrossRef]22. Argyros, I.K. Convergence and Application of Newton-Type Iterations; Springer: Berlin/Heidelberg, Germany, 2008.23. Petkovic, M.S.; Neta, B.; Petkovic, L.D.; Dzunic, J. Multipoint Methods for Solving Nonlinear Equations; Academic

Press: Amsterdam, The Netherlands, 2012.24. Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964.

373


25. Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. A modified Newton-Jarratt’s composition. Numer. Algor.2010, 55, 87–99. [CrossRef]

26. Abbasbandy, S.; Bakhtiari, P.; Cordero, A.; Torregrosa, J.R.; Lotfi, T. New efficient methods for solving nonlinearsystems of equations with arbitrary even order. Appl. Math. Comput. 2016, 287, 287–288. [CrossRef]

27. Hueso, J.L.; Martínez, E.; Teruel, C. Convergence, efficiency and dynamics of new fourth and sixth orderfamilies of iterative methods for nonlinear systems. J. Comput. Appl. Math. 2015, 275, 412–420. [CrossRef]

28. Sharma, J.R.; Arora, H. Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 2014,51, 193–210. [CrossRef]

29. Wang, X.; Li, Y. An Efficient Sixth Order Newton Type Method for Solving Nonlinear Systems. Algorithms 2017,10, 45. [CrossRef]

30. Burden, R.L.; Faires, J.D. Numerical Analysis; PWS Publishing Company: Boston, MA, USA, 2001.31. Simpson, R.B. A method for the numerical determination of bifurcation states of nonlinear systems of equations.

SIAM J. Numer. Anal. 1975, 12, 439–451. [CrossRef]32. Kapania, R.K. A pseudo-spectral solution of 2-parameter Bratu’s equation. Comput. Mech. 1990, 6, 55–63.

[CrossRef]33. Sauer, T. Numerical Analysis, 2nd ed.; Pearson: Upper Saddle River, NJ, USA, 2012.

c© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution (CCBY) license (http://creativecommons.org/licenses/by/4.0/).

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mathematics

Article

A Seventh-Order Scheme for Computing theGeneralized Drazin Inverse

Dilan Ahmed 1 , Mudhafar Hama 2, Karwan Hama Faraj Jwamer 2 and Stanford Shateyi 3,*

1 Department of Mathematics, College of Education, University of Sulaimani, Kurdistan Region,Sulaimani 46001, Iraq

2 Department of Mathematics, College of Science, University of Sulaimani, Kurdistan Region,Sulaimani 46001, Iraq

3 Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa* Correspondence: [email protected]

Received: 26 May 2019; Accepted: 22 June 2019; Published: 12 July 2019

Abstract: One of the most important generalized inverses is the Drazin inverse, which is definedfor square matrices having an index. The objective of this work is to investigate and present acomputational tool in the form of an iterative method for computing this task. This scheme reachesthe seventh rate of convergence as long as a suitable initial matrix is chosen and by employing onlyfive matrix products per cycle. After some analytical discussions, several tests are provided to showthe efficiency of the presented formulation.

Keywords: drazin inverse; generalized inverse; iterative methods; higher order; efficiency index

JEL Classification: 15A09; 65F30.

1. Introduction

Drazin, in the pioneering work in [1], proposed and generalized a different type of outer inversein associative rings and semigroups that does not possess the reflexivity feature but commutes withthe element. The significance of this type of inverse and its calculation was then discussed whollyin [2]. Accordingly, several authors attempted to propose procedures for the calculation of generalizedinverses. See, e.g., [3–5].

It is recalled that the smallest non-negative integer k such that [6]

rank(Ak) = rank(Ak+1), (1)

is named as the matrix A’s index and is shown by ind(A). Furthermore, assume that A is an N × Nmatrix with complex entries. The Drazin inverse of A, shown by AD, is the unique matrix X readingthe following identities [7]:

1. AkXA = Ak,2. XAX = X,3. AX = XA.

Throughout the paper, with A∗, R(A), N (A), and rank(A), we show the conjugate transpose,the range, the null space, and the rank of A ∈ CN×N , respectively [8]. It is remarked that if ind(A) = 1,X is named as the g-inverse or group inverse of A. In addition, if A is nonsingular, then it is easilyseen that

indA = 0, and AD = A−1. (2)



For the square system Ax = b, the general solution can be represented using the concept of theDrazin inverse as follows [7]:

x = ADb + (I − AAD)z, (3)

wherein z ∈ R(Ak−1) +N (A).Methods in the category of iteration schemes for computing generalized inverses like the Drazin

inverse are quite sensitive to the choice of the initial approximation. In fact, the convergence of theSchulz-type method can only be observed if the initial value is chosen correctly [9,10]. This selectioncan be done under special care on some criteria, as already discussed, and given in the literature fordifferent types of outer inverses. For some in-depth discussions about this, one may refer to [11].

Authors in [12] showed that iterative Schulz-type schemes can be used for finding the Drazininverse of square matrices both possessing real or complex spectra. Actually, the authors investigatedthe initial matrix below:

X0 = αAl , l ≥ ind(A) = k, (4)

wherein the parameter α should be selected such that the criterion ‖I − AX0‖ < 1 is read. Employingthe starting value (4) yields an iterative scheme for computing the famous Drazin inverse withsecond-order convergence.

It is in fact necessary to apply an appropriate initial matrix when calculating the Drazin inverse.One way is as follows [12,13]:

X0 =2

Tr(Ak+1)Ak, (5)

wherein Tr(·) stands for the trace of an arbitrary square matrix. Another fruitful initial matrix whichcould lead in converging sequence of matrices for computing the generalized Drazin inverse can bewritten as

X0 =1

2‖A‖k+12

Ak. (6)

The Schulz method of the quadratic convergence rate for doing this task can be defined by [14]

Xn+1 = Xn(2I − AXn), n = 0, 1, 2, · · · , (7)

where I is the identity matrix and requires only two matrix products to achieve this rate per cycle.Authors in [15] re-studied Chebyshev’s method for calculating A†

MN using a suitable initial valueas follows:

Xn+1 = Xn(3I − AXn(3I − AXn)), n = 0, 1, 2, · · · , (8)

with a third-order of convergence having three matrix products per cycle. Another scheme, having acubic rate of convergence and a greater number of matrix products, was given by the same authorsas follows:

Xn+1 = Xn

[I +

12(I − AXn)(I + (2I − AXn)

2)

], n = 0, 1, 2, · · · . (9)

A general procedure for having p-order methods with a p number of matrix–matrix products wasgiven in [16] (Chapter 5). For instance, the authors presented the following fourth-order scheme:

Xn+1 = Xn(I + Bn(I + Bn(I + Bn))), n = 0, 1, 2, · · · , (10)

in which Bn = I − AXn.The main goal and motivation for investigating novel or useful matrix schemes for computing the

Drazin inverse is not only because of the applications of such solvers in different kinds of mathematicalproblems [17,18] but also to improve the computational efficiency index. In fact, the hyperpowerstructure as given for a sample case in (10) requires a p number of matrix–matrix products to achieve a

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p rate of convergence. This leads to more and more inefficient methods from this class of methods,particularly when the order increases.

As such, motivated by extending efficient methods of higher orders for calculating generalizedinverses, here we focus on a seventh-order scheme and discuss how we can reach this higher rateby employing only five matrix by matrix products. This will reveal a higher efficiency index for thediscussed scheme in computing the Drazin inverse. For more studies and investigations in this fieldand related issues of generalized matrix inverses, interested readers are guided to [19–22].

The organization of this paper is as follows. In Section 1, we quickly review the definition,literature, and the need for development of higher order schemes. Section 2 is devoted to extendingand proposing an efficient iterative expression for the Drazin inverse. It is derived that the schemerequires only five matrix–matrix products to achieve this rate.

Theoretical discussions with some concrete proofs are provided in Section 3, while Section 4 isoriented on the application of this scheme for computing the Drazin inverses. Results will reveal theeffectiveness of this scheme for calculating the Drazin inverse. Lastly, some concluding comments aregiven in Section 5.

2. Derivation of an Efficient Formulation

Here, the aim is to present a competitive formulation for a member of the hyperpower familyof iterations, so as to not only gain a high rate of convergence but also improve the computationalefficiency index. In fact, we must factorize the formulation so as to gain the same high convergencerate but a lower number of matrix products.

Toward this objective, let us take into account the following seventh-order method from the familyof hyperpower iteration schemes [23]:

Xn+1 = Xn(I + Bn + B2n + B3

n + B4n + B5

n + B6n). (11)

It is necessary to emphasize that we are looking for a seventh-order scheme and not ahigher-order one, since we wish to hit several targets at the same time. First, the derived scheme forthe Drazin inverse must be efficient, viz., it must improve the computational efficiency index of theexisting solvers, as will be shown at the end of this section. Second, very high-order schemes mightoccasionally become hard for coding purposes, and this limits their application, so we aim to havethis order be high but not so high. Besides, higher-order schemes mean that fewer stopping criteria(the computation of matrix norms) should be calculated per cycle, which is useful in terms of theelapsed time.

Now, to improve the performance of (11), we factorize (11) in what follows:

Xn+1 = Xn

(I + Bn(I + Bn)(I − Bn + B2

n)(I + Bn + B2n))

. (12)

However, the formulation (12) needs six matrix–matrix multiplications, which is still not thatuseful for improving the computational index of efficiency theoretically. As such, doing morefactorization would yield the following scheme:

Xn+1 = Xn

(I + (Bn + B2

n)(I − Bn + B2n)(I + Bn + B2

n))

. (13)

The scheme (13) requires five matrix products per cycle to achieve the seventh order ofconvergence. Noting that one reason for the need to propose and have an efficient higher scheme inthe category of matrix Schulz-type methods is also in the fact that Schulz-type schemes of lower ordersare quite slow at the initial stage of iterates, and this could yield a greater computational burden forfinding the Drazin inverse [24]. In fact, it sometimes takes several iterates for the scheme to arrive atits convergence phase, and due to imposing some stopping termination based on matrix norms, thismight add some more elapsed time for the application of lower-order schemes.

377


It is recalled that the definition of index of efficiency is given by [25]:

EI = ρ1κ , (14)

wherein ρ and κ are the convergence rate and the total cost per cycle, respectively.Hence, the efficiency indexes of different methods are reported by

EI(7) = 212 � 1.41421, EI(8) = 3

13 � 1.44225, (15)

EI(9) = 414 � 1.41421, EI(12) = 7

16 � 1.38309, EI(13) = 7

15 � 1.47577. (16)

This shows that we have achieved our motivation by improving the efficiency index for calculatingthe Drazin inverse via a competitive formulation.

3. Seventh Rate of Convergence

Let us now recall some of the well-known lemmas we need in the rest of this section.

Proposition 1 ([26]). Assume that M ∈ Cn×n and ε > 0 are given. There is at least one matrix norm ‖ · ‖such that

ρ(M) ≤ ‖M‖ ≤ ρ(M) + ε, (17)

wherein ρ(M) shows the collection of all of M’s eigenvalues (in the maximum of absolute value sense).

Proposition 2 ([26]). If PL,M shows the projector on a space L on space M, then(i) PL,MQ = Q if and only ifR(Q) ⊆ L,(ii) QPL,M = Q if and only if N (Q) ⊇ M.

The proof of the main theorem concerning the convergence as well as its rate of (13) for calculatingthe generalized Drazin inverse is now addressed as follows.

Theorem 1. Consider that A ∈ CN×N is a square singular matrix. In addition, let the initial value X0 beselected via (4) or (5). Thence, the matrices {Xn}n=∞

n=0 generated via (13) satisfy the following error estimate forcalculating the Drazin inverse:

‖AD − Xn‖ ≤ ‖AD‖‖I − AX0‖7n. (18)

In addition, the convergence order is seven.

Proof. To prove that the sequence is converging, we first take into consideration that

Rn+1 = I − AXn+1

= I − A(Xn

(I + (Bn + B2

n)(I − Bn + B2n)(I + Bn + B2

n)))

= I − A(Xn

(I + Bn(I + Bn)(I − Bn + B2

n)(I + Bn + B2n)))

= I − A(Xn

(I + Bn + B2

n + B3n + B4

n + B5n + B6

n

))

= (I − AXn)7

= R7n,

(19)

wherein Rn = I − AXn. Employing a matrix norm on (19), we obtain that

‖Rn+1‖ ≤ ‖Rn‖7. (20)

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Since X0 is selected as in (4) or (5), we have

R(X0) ⊆ R(Ak). (21)

This could now be stated asR(Xn) ⊆ R(Xn−1). (22)

Thus, we can conclude thatR(Xn) ⊆ R(Ak), n ≥ 0. (23)

In a similar way, by defining the scheme by the left-multiplying of Xn, we can state that

N (Xn) ⊇ N (Ak), n ≥ 0. (24)

Now, an application of the definition of the Drazin inverse yields

AAD = AD A = PR(Ak),N (Ak). (25)

Proposition 2, along with (23), (24), and (25) could lead to

Xn AAD = Xn = AD AXn, n ≥ 0. (26)

To complete the proof, we proceed in what follows. The error matrix δn = AD − Xn satisfies

δn = AD − Xn

= AD − AD AXn

= AD (I − AXn)

= ADRn.

(27)

Using (20), we obtain the following inequality

‖δn‖ = ‖AD‖‖Rn‖ ≤ ‖AD‖‖R0‖7n, (28)

which is an affirmation of (18). Employing (28) and Proposition 2, one gets that

Aδn+1 = AAD − AXn+1

= AAD − I + I − AXn+1

= AAD − I + Rn+1.

(29)

Note that the idempotent matrix AAD is the projector on R(Ak) along N (Ak), where R(Ak)

denotes the range of Ak, and N (Ak) is the null space of Ak. Considering (19) and applying severalsimplifications, one obtains

Aδn+1 = AAD − I + R7n. (30)

Now, by taking into account the following feature,

(I − AAD)t = (I − AAD), t ≥ 1, (31)

we can get that(I − AAD)Aδn = (I − AAD)A(AD − Xn)

= Xn − AADVn

= 0.

(32)

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We obtain, for each t ≥ 1, that (here we use (32) in simplifications)

(Rn)t + AAD − I = (I − AVn)

t + AAD − I

= (I − AAD + AAD − AVn)t + AAD − I

=((I − AAD) + Aδn

)t+ AAD − I

= I − AAD + (Aδn)t + AAD − I

= (Aδn)t.

(33)

From (33) and (30), we haveAδn+1 = (Aδn)

7. (34)

Taking matrix norms from both sides yields

‖Aδn+1‖ ≤ ‖Aδn‖7. (35)

Considering (35) and the second criterion of (1), we obtain that

‖δn+1‖ = ‖Xn+1 − AD‖= ‖AD AVn+1 − AD AAD‖= ‖AD(AVn+1 − AAD)‖≤ ‖AD‖‖Aδn+1‖≤ ‖AD‖‖δn‖7.

(36)

The relations in (36) yield the point that Xn → AD as n → +∞ with the seventh order ofconvergence. The proof is complete.


The purpose of this section is to investigate the efficiency of our competitive formulation forcomputing the Drazin inverse, both theoretically and numerically. For such a task, we employ thechallenging schemes (7), (8), and (13).

Here, we have simulated the tests in Mathematica 11.0, [27,28] and the time shown is in seconds.Noting that the compared methods are programmed in the same environment using the hardwareCPU Intel Core i5 2430–M, 16 GB RAM in Windows 7 Ultimate with an SSD hard disk.

Test Problem 1. The aim of this test is to testify the computation of the Drazin inverse for the followinginput [12]:

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 0.4 0 0 0 0 0 0 0 0 0 0−2 0.4 0 0 0 0 0 0 0 0 0 0−1 −1 1 −1 0 0 0 0 −1 0 0 0−1 −1 −1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 −1 −1 0 0 −1 00 0 0 0 1 1 −1 −1 0 0 0 00 0 0 −1 −2 0.4 0 0 0 0 0 00 0 0 0 2 0.4 0 0 0 0 0 00 −1 0 0 0 0 0 0 1 −1 −1 −10 0 0 0 0 0 0 0 −1 1 −1 −10 0 0 0 0 0 0 0 0 0 0.4 −20 0 0 0 0 0 0 0 0 0 0.4 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

380


at which k = ind(A) = 3. Here, the Drazin inverse is expressed by

AD =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.25 −0.25 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.1.25 1.25 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

−1.66406 −0.992187 0.25 −0.25 0. 0. 0. 0. −0.0625 −0.0625 0. 0.15625−1.19531 −0.679687 −0.25 0.25 0. 0. 0. 0. −0.0625 0.1875 0.6875 1.34375−2.76367 −1.04492 −1.875 −1.25 −1.25 1.25 1.25 1.25 1.48438 2.57813 3.32031 6.64063−2.76367 −1.04492 −1.875 −1.25 −1.25 1.25 1.25 1.25 1.48438 2.57813 4.57031 8.5156314.1094 6.30078 6.625 3.375 5. −3. −5. −5. −4.1875 −8.5 −10.5078 −22.4609−19.3242 −8.50781 −9.75 −5.25 −7.5 4.5 7.5 7.5 6.375 12.5625 15.9766 33.7891−0.625 −0.3125 0. 0. 0. 0. 0. 0. 0.25 −0.25 −0.875 −1.625−1.25 −0.9375 0. 0. 0. 0. 0. 0. −0.25 0.25 −0.875 −1.625

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.25 1.250. 0. 0. 0. 0. 0. 0. 0. 0. 0. −0.25 0.25

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

The results are obtained by applying the stop termination

||Xn+1 − Xn||1 ≤ 10−6, (37)

and now by employing the definition in Section 1, we have for (13),

‖Ak+1Xn+1 − Ak‖∞ � 3.69638× 10−12,

‖Xn+1 AXn+1 − Xn+1‖∞ � 8.43992× 10−10,

‖AXn+1 − Xn+1 A‖∞ � 3.75205× 10−10.

(38)

It is also necessary to mention that the domain of validity for the proposed formulation (13)is not only limited to the Drazin inverse, and if a suitable initial approximation is used, undersome assumptions we can construct a converging sequence of matrix iterates for other types ofgeneralized inverses.

Test Problem 2. In this test, we compare the results of various schemes for computing the regular inverse usingthe initial matrix

X0 =1

‖A‖FA∗, (39)

the stopping condition (37), and the following complex matrices constructed in Mathematica:

N = 5000; no = 25;

ParallelTable[

A[j] = SparseArray[

{Band[{-100, 1100}] -> RandomReal[20], Band[{1, 1}] -> 2.,

Band[{1000, -50}, {N - 20, N - 25}] -> {2.8, RandomReal[] + I},

Band[{600, 150}, {N - 100, N - 400}] -> {-RandomReal[3], 3. + 3 I}

},

{N, N}, 0.],

{j, no}

];

The plot of the large sparse matrices in Test Problem 2 is plotted in Figure 1, showing the sparsitypattern of these matrices, while the pattern of sparsity for the inverse matrix is provided in Figure 2.Figure 3 shows the clear superiority of the proposed formulation in computing the inverse of largesparse matrices.

Here, we give a simple Mathematica implementation of (13) in solving Test Problem 2:

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For[j = 1, j <= number, j++,

{

X = A[j]/(Norm[A[j], "Frobenius"]^2);

k = 1;

X1 = 20 X;

Time[j] = Part[

While[k <= 75 && N[Norm[X - X1, 1]] >= 10^(-6),

X1 = SparseArray[X];

XX = Id - A[j].X1;

X2 = XX.XX;

X =

Chop@

SparseArray[

X1.(Id + (XX + X2).(Id - XX + X2).(Id + XX + X2))];

k++]; // AbsoluteTiming,

1];

}];

To apply our scheme in modern applications of numerical linear algebra getting involved withsparse large matrices, one may use some commands such as SparseArray[] for tackling matrices insparse forms and subsequently reduce the computational effort and time for preserving the sparsitypattern and finding an approximate inverse. Such applications may occur in various types of problemslike the ones in [29,30].

Figure 1. The sparsity pattern of matrices in Test Problem 2.

Figure 2. The sparsity pattern of the inverse matrix X = A−125 in Test Problem 2.

382


Figure 3. The CPU time required for different matrices in Test Problem 2.

5. Conclusions

Following the motivation of proposing and extending efficient iteration schemes for computinggeneralized inverses, and particularly for Drazin inverses, in this work we have extended and discussedtheoretically how we could achieve a seventh rate in a hyperpower structure for an iterative method.The scheme is a matrix product method and employs only five products to reach this rate. Clearly, theefficiency index will hit the bound 71/5 � 1.47577.

Several computational tests for calculating the Drazin inverses of several randomly generatedmatrices were provided to show the superiority and stability of the scheme in doing this task. Othercomputational problems of different sizes for different matrices were also done and showed similarbehavior and the superiority of (13) for the Drazin inverse.

Author Contributions: All authors contributed equally in preparing and writing this work.

Funding: This manuscript receives no funding.

Acknowledgments: We are grateful to four anonymous referees for several comments which improved thereadability of this work.


References

1. Drazin, M.P. Pseudoinverses in associative rings and semigroups. Am. Math. Mon. 1958, 65, 506–514. [CrossRef]2. Wilkinson, J.H. Note on the Practical Significance of the Drazin Inverse; Campbell, S.L., Ed.; Recent Applications

of Generalized Inverses, Pitman Advanced Publishing Program, Research Notes in Mathematics, No. 66,Boston; NASA: Washington, DC, USA, 1982; pp. 82–99.

3. Kyrchei, I. Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrixand differential matrix equations. Appl. Math. Comput. 2013, 219, 7632–7644. [CrossRef]

4. Liu, X.; Zhu, G.; Zhou, G.; Yu, Y. An analog of the adjugate matrix for the outer inverse A(2)T,S. Math. Prob. Eng.

2012, 2012, 591256.5. Moghani, Z.N.; Khanehgir, M.; Karizaki, M.M. Explicit solution to the operator equation AXD + FX∗B = C

over Hilbert C∗-modules. J. Math. Anal. 2019, 10, 52–64.6. Ben-Israel, A.; Greville, T.N.E. Generalized Inverses: Theory and Applications, 2nd ed.; Springer: New York, NY,

USA, 2003.7. Wei, Y. Index splitting for the Drazin inverse and the singular linear system. Appl. Math. Comput. 1998, 95,

115–124. [CrossRef]8. Ma, H.; Li, N.; Stanimirovic, P.S.; Katsikis, V.N. Perturbation theory for Moore–Penrose inverse of tensor via

Einstein product. Comput. Appl. Math. 2019, 38, 111. [CrossRef]9. Soleimani, F.; Soleymani, F.; Shateyi, S. Some iterative methods free from derivatives and their basins of

attraction. Discret. Dyn. Nat. Soc. 2013, 2013, 301718. [CrossRef]

383


10. Soleymani, F. Efficient optimal eighth-order derivative-free methods for nonlinear equations.Jpn. J. Ind. Appl. Math. 2013, 30, 287–306. [CrossRef]

11. Pan, V.Y. Structured Matrices and Polynomials: Unified Superfast Algorithms; BirkhWauser: Boston, MA, USA;Springer: New York, NY, USA, 2001.

12. Li, X.; Wei, Y. Iterative methods for the Drazin inverse of a matrix with a complex spectrum.Appl. Math. Comput. 2004, 147, 855–862. [CrossRef]

13. Stanimirovic, P.S.; Ciric, M.; Stojanovic, I.; Gerontitis, D. Conditions for existence, representations, andcomputation of matrix generalized inverses. Complexity 2017, 2017, 6429725. [CrossRef]

14. Schulz, G. Iterative Berechnung der Reziproken matrix. Z. Angew. Math. Mech. 1933, 13, 57–59. [CrossRef]15. Li, H.-B.; Huang, T.-Z.; Zhang, Y.; Liu, X.-P.; Gu, T.-X. Chebyshev-type methods and preconditioning

techniques. Appl. Math. Comput. 2011, 218, 260–270. [CrossRef]16. Krishnamurthy, E.V.; Sen, S.K. Numerical Algorithms: Computations in Science and Engineering; Affiliated

East-West Press: New Delhi, India, 1986.17. Ma, J.; Gao, F.; Li, Y. An efficient method to compute different types of generalized inverses based on linear

transformation. Appl. Math. Comput. 2019, 349, 367–380. [CrossRef]18. Soleymani, F.; Stanimirovic, P.S.; Khaksar Haghani, F. On Hyperpower family of iterations for computing

outer inverses possessing high efficiencies. Linear Algebra Appl. 2015, 484, 477–495. [CrossRef]19. Qin, Y.; Liu, X.; Benítez, J. Some results on the symmetric representation of the generalized Drazin inverse in

a Banach algebra. Symmetry 2019, 11, 105. [CrossRef]20. Wang, G.; Wei, Y.; Qiao, S. Generalized Inverses: Theory and Computations; Science Press: Beijing, China;

New York, NY, USA, 2004.21. Xiong, Z.; Liu, Z. The forward order law for least Squareg-inverse of multiple matrix products. Mathematics

2019, 7, 277. [CrossRef]22. Zhao, L. The expression of the Drazin Inverse with rank constraints. J. Appl. Math. 2012, 2012, 390592. [CrossRef]23. Sen, S.K.; Prabhu, S.S. Optimal iterative schemes for computing Moore-Penrose matrix inverse. Int. J.

Syst. Sci. 1976, 8, 748–753. [CrossRef]24. Soleymani, F. An efficient and stable Newton–type iterative method for computing generalized inverse A(2)

T,S.Numer. Algorithms 2015, 69, 569–578. [CrossRef]

25. Ostrowski, A.M. Sur quelques transformations de la serie de LiouvilleNewman. C.R. Acad. Sci. Paris 1938,206, 1345–1347.

26. Jebreen, H.B.; Chalco-Cano, Y. An improved computationally efficient method for finding the Drazin inverse.Discret. Dyn. Nat. Soc. 2018, 2018, 6758302. [CrossRef]

27. Sánchez León, J.G. Mathematica Beyond Mathematics: The Wolfram Language in the Real World; Taylor & FrancisGroup: Boca Raton, FL, USA, 2017.

28. Wagon, S. Mathematica in Action, 3rd ed.; Springer: Berlin, Germany, 2010.29. Soleymani, F. Efficient semi-discretization techniques for pricing European and American basket options.

Comput. Econ. 2019, 53, 1487–1508. [CrossRef]30. Soleymani, F.; Barfeie, M. Pricing options under stochastic volatility jump model: A stable adaptive scheme.

Appl. Numer. Math. 2019, 145, 69–89. [CrossRef]


384

mathematics

Article

Calculating the Weighted Moore–Penrose Inverse bya High Order Iteration Scheme

Haifa Bin Jebreen

Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia;[email protected]

Received: 16 July 2019; Accepted: 7 August 2019; Published: 10 August 2019

Abstract: The goal of this research is to extend and investigate an improved approach for calculatingthe weighted Moore–Penrose (WMP) inverses of singular or rectangular matrices. The scheme isconstructed based on a hyperpower method of order ten. It is shown that the improved schemeconverges with this rate using only six matrix products per cycle. Several tests are conducted to revealthe applicability and efficiency of the discussed method, in contrast with its well-known competitors.

Keywords: iteration scheme; Moore–Penrose; rectangular matrices; rate of convergence;efficiency index

MSC: 15A09; 65F30

1. Introduction

1.1. Background

Constructing and discussing different features of iterative schemes for the calculation of outerinverses is an active topic of current research in Applied Mathematics (for more details, refer to [1–3]).Many papers have been published in the field of outer inverses over the past few decades, each havingtheir own domain of validity and usefulness. In fact, in 1920, Moore was a pioneer of this field andpublished seminal works about the outer inverse [4,5]. However, several deep works were publishedduring the 1950s (as reviewed and observed in [4]). It is also noted that pseudo-inverse operator wasfirst introduced by Fredholm in [6].

The method of partitioning (due to Greville) was a pioneering work in computing generalizedinverses, which was re-introduced and re-investigated in [4,7]. This scheme requires a lot of operationsand is subject to cancelation and rounding errors. Among the generalized inverses, the weightedMoore–Penrose (WMP) inverse is important, as it can be simplified to a pseudo-inverse, as well asa regular inverse. Several applications of computing the WMP inverse can be observed, with somediscussion, in the recent literature [8,9]; including applications to the solution of matrix equations.See [10–13] for further discussions and applications.

Furthermore, for large matrices, or as long as the weight matrices in the process of computing theWMP inverse are ill-conditioned, symbolic computation of the current algorithms may not properlywork due to several reasons, such as time consumption, requiring higher memory space, or instability.On the other hand, several numerical methods for the weighted Moore–Penrose (WMP) inverse arenot stabie or possess slow convergence rates. Hence, it is necessary to investigate and extend noveland useful iterative matrix methods for such an objective; see, also, the discussions in [14,15].



1.2. Definition

Let us consider that M and N are two square Hermitian positive definite (HPD) matrices ofsizes m and n (m ≤ n) and A ∈ Cm×n. Then, there is a unique matrix X satisfying the followingidentities [16]:

1. AXA = A,2. XAX = X,3. (MAX)∗ = MAX,4. (NXA)∗ = NXA.

Then, X ∈ Cn×m is called the WMP inverse of A, and is shown by A†MN . Noting that, as long as

M = Im×m and N = In×n, then X is the Moore–Penrose (MP) inverse, or simply the pseudo-inverse ofA, and we show it by A† [17]. Furthermore, when the matrix A is non-singular, then the pseudo-inversewill be simplified to the regular inverse.

The weighted singular value decomposition (WSVD), first introduced in [18], is normally appliedto define this generalized inverse. Consider that the rank of A is r. Then, we have U ∈ Cm×m andV ∈ Cn×n, satisfying the following relations:

U∗MU = Im×m, (1)

andV∗N−1V = In×n, (2)

such that

A = U

(D 00 0

)V∗. (3)

Thus, A†MN is furnished as follows:

A†MN = N−1V

(D−1 0

0 0

)U∗M, (4)

where we have a diagonal matrix D = diag(σ1, σ2, . . . , σr), for σ1 ≥ σ2 ≥ . . . ≥ σr > 0, while σ2i is the

non-zero eigenvalue of N−1 A∗MA. In addition,

‖A‖MN = σ1, ‖A†MN‖NM =

1σr

. (5)

In this work, A# = N−1 A∗M is used as the weighted matrix of the conjugate transpose of A.See [19] for more details.

1.3. Literature

Schulz-type methods for the calculation of the WMP inverse are sensitive to the choice of theinitial value; that is, the initial choice of matrix must be close enough to the generalized inverse so asto guarantee the scheme to converge [20]. More precisely, convergence can only be observed if thestarting matrix is chosen carefully. However, this starting value can be chosen simply for the case ofthe WMP inverse. The pioneering work in [21] gave several suggestions, along with deep discussions,about how to make such a choice quickly.

Let us, now, briefly provide some of the pioneering and most important matrix iterative methodsfor computing the WMP inverse.

386


The second–order Schulz scheme for finding the WMP inverse, requiring only two matrix productsper computing cycle, is given by [22]:

Xk+1 = Xk(2I − AXk), k = 0, 1, 2, · · · . (6)

Throughout the work, I stands for the identity matrix, unless clearly stated otherwise.An improvement of (6) with third-order convergence, known as Chebyshev’s method, was

discussed in [23] for computing A†MN as follows:

Xk+1 = Xk(3I − AXk(3I − AXk)), k = 0, 1, 2, · · · . (7)

The authors in [23] proposed another third-order iterative formulation, having one more matrixmultiplication, as follows:

Xk+1 = Xk

[I +

12(I − AXk)(I + (2I − AXk)

2)

], k = 0, 1, 2, · · · . (8)

It is necessary to recall that a general class of iteration schemes for computing the WMPinverse and some other kinds of other generalized inverses was discussed and investigated in [24](Chapter 5) to have p-th order using a total of p matrix products. An example could be the followingfourth-order iteration:

Xk+1 = Xk(I + Bk(I + Bk(I + Bk))), k = 0, 1, 2, · · · , (9)

where Bk = I − AXk. As another instance, a tenth-order matrix method could be furnished asfollows [25]:

Xk+1 = Xk(I + Bk(I + Bk(I + Bk(I + Bk(I + Bk(I+Bk(I + Bk(I + Bk(I + Bk))))))))), k = 0, 1, 2, · · · .

(10)

1.4. Motivation and Organization

The main motivation behind proposing and extending new iterative methods for the WMPinverse is to apply them in practical large scale problems [26], as well as to improve the computationalefficiency, which is directly linked to the concept of numerical analysis for designing new iterativeexpressions which are economically useful, in being able to reduce computational complexity andtime requirements.

Hence, with this motivation at hand, to increase the computational efficiency index as well as tocontribute in this field, the main focus of this work is to investigate a tenth-order method requiringonly six matrix multiplications per cycle. We prove that this can provide an improvement of thecomputational efficiency index in calculating the WMP inverse.

The paper is organized as follows. Section 1 discusses the preliminaries and literature of this topicvery briefly, to prepare the reader for the analytical discussions of Section 2, in which we describe aneffective iteration formulation for the WMP inverse. It is investigated that the method needs only sixmatrix multiplications to reach its tenth order of convergence.

Concrete proofs of convergence are furnished in Section 3. Section 4 discusses the application ofour formulation to the WMP inverses of many randomly generated matrices of various dimensions.Numerical evidence demonstrates the usefulness of this method for computing the WMP inverse, interms of the elapsed computation time. Finally, several concluding remarks and comments are givenin Section 5.

387


2. A High Order Scheme for the WMP Inverse

For the use of iterative methods, such as the ones described in Section 1, it is required to employa starting value when computing the WMP inverse. As in [27], one general procedure to find thisstarting matrix is of the following form:

X0 = λA#, (11)

where A# = N−1 A∗M is the matrix of weighted conjugate transpose (WCT) for the input matrix A and

λ =1σ2

1. (12)

Recall that, in (12), σ1 is the the largest eigenvalue of N−1 A∗MA.

2.1. Derivation

Another reason for proposing a higher order method is that methods based on improvements ofthe Schulz iteration scheme are slow in the initial phase of iteration. This means that the convergenceorder cannot be observed at the beginning, it can be seen only after performing several iterates. On theother hand, by incorporating a stop condition using matrix norms, we can increase the elapsed time ofexecuting the written programs for finding the WMP inverse.

Accordingly, to contribute and extend a high order matrix iteration scheme in this context, we firsttake into account a tenth-order scheme having ten matrix multiplications per cycle, as follows:

Xk+1 = Xk(I + Bk + B2k + · · ·+ B9

k). (13)

Now, to develop the performance of (13), we factorize (13) to reduce the number of products. So,we can write

Xk+1 = Xk (I + Bk) [(I − Bk + B2k − B3

k + B4k)(I + Bk + B2

k + B3k + B4

k)]. (14)

This formulation for the matrix iteration requires seven matrix products. However, it is possibleto reduce this number of products by considering a more tight formulation for (14). Hence, we write

Xk+1 = Xk (I + Bk) Mk, (15)

whereMk = [(I + χB2

k + B4k)(I + κB2

k + B4k)]. (16)

To find the unknown weighting coefficients in (15) and, more specifically, in (16), we need to solvea symbolic problem. As such, a Mathematica code [28] was employed to do such a task, as follows:

ClearAll["Global‘*"];

fact1 = (1 + a B^2 + B^4);

fact2 = (1 + b B^2 + B^4);

sol = fact1*fact2 + (c B^2) // Expand

S = Table[

s[i] = Coefficient[sol, B^i], {i, 2, 6, 2}

] // Simplify

Solve[

s[2] == 1 && s[4] == 1 && s[6] == 1, {a, b, c}

] // Simplify

{a, b, c} = {a, b, c} /. %[[1]] // Simplify

Chop@sol // Simplify

388


This was given only to ease understanding of the procedure of obtaining the coefficient. Now,we obtain:

χ =12

(1−

√5)

, κ =12

(1 +

√5)

. (17)

This means that (15) requires only six matrix products per cycle to hit a convergence speed of ten.

2.2. Several Lemmas

Before providing the main results concerning the convergence analysis of the proposed scheme,we furnish the following lemmas, inspired by [29], which reveal how the iterates generated by (15)have some specific important relations and, then, show a relation between (4) and (15).

Lemma 1. For {Xk}k=∞k=0 produced by (15) using the starting matrix (11), for any k ≥ 0, it holds that

(MAXk)∗ = MAXk,

(NXk A)∗ = NXk A,Xk AA†

MN = Xk,A†

MN AXk = Xk.

(18)

Proof. The proof can be done by employing mathematical induction. When k = 0 and X0 is thesuitable initial matrix, the first two relations in (18) are straightforward. Hence, we discuss the last tworelations by applying the following identities:

(AA†MN)

# = AA†MN , (19)

and(A†

MN A)# = A†MN A. (20)

Accordingly, we have:X0 AA†

MN = λA# AA†MN

= λA#(AA†MN)

#

= λA#(A†MN)

# A#

= λ(AA†MN A)#

= λA#

= X0,

(21)

and alsoA†

MN AX0 = λA†MN AA#

= λ(A†MN A)# A#

= λ(A#(A†MN)

# A#)

= λ(A(A†MN A)#

= λA#

= X0.

(22)

Subsequently, now the relation is valid for k > 0, then we discuss that it will still be true for k + 1.Taking our matrix iteration (15) into consideration, we have:

(MAXk+1)∗ = (MA(Xk (I + Bk) [(I + χB2

k + B4k)(I + κB2

k + B4k)]))

∗

= [MAXk(

I + Bk + B2k + B3

k + B4k + B5

k + B6k + B7

k + B8k + B9

k)]∗

= MA[Xk(

I + Bk + B2k + B3

k + B4k + B5

k + B6k + B7

k + B8k + B9

k)]

= MAXk+1,

using that(M(AXk))

∗ = MAXk, (23)

389


M is a Hermitian positive definite matrix (M∗ = M), and similar facts, such as:

(M(AXk)2)∗ = (M(AXk)(AXk))

∗

= (AXk)∗(M(AXk))

∗

= (AXk)∗(M(AXk))

= (AXk)∗M∗(AXk)

= (M(AXk))∗(AXk)

= M(AXk)(AXk)

= M(AXk)2.

(24)

Hence, the first relation in (18) is true for k + 1, and the 2nd relation could be investigated similarly.For the other relation in (18), by employing the assumption that

Xk AA†MN = Xk, (25)

and (15), we have:

Xk+1 AA†MN = (Xk (I + Bk) [(I + χB2

k + B4k)(I + κB2

k + B4k)])AA†

MN= (Xk + XkBk + XkB2

k + XkB3k + XkB4

k + XkB5k + XkB6

k+XkB7

k + XkB8k + XkB9

k)AA†MN

= Xk AA†MN + XkBk AA†

MN + XkB2k AA†

MN + XkB3k AA†

MN + XkB4k AA†

MN+XkB5

k AA†MN + XkB6

k AA†MN + XkB7

k AA†MN + XkB8

k AA†MN + XkB9

k AA†MN

= (Xk + XkBk + XkB2k + XkB3

k + XkB4k + XkB5

k+XkB6

k + XkB7k + XkB8

k + XkB9k)

= Xk+1.

Therefore, the third relation in (18) is valid for k + 1. The final relation could be investigated in asimilar way, and the result follows. The proof is, thus, complete.

Lemma 2. Employing the assumptions of Lemma 1 and (3), then for (15) we have:

(V−1N)Xk(M−1(U∗)−1) = diag(Tk, 0), (26)

where Tk is a diagonal matrix, V∗N−1V = In×n, U∗MU = Im×m, V ∈ Cn×n, U ∈ Cm×m, and A = UΣV∗.

Proof. Assume that T0 = λD and that σ2i are the non-zero eigenvalues of the matrix N−1 A∗MA, while

D = diag(σ1, σ2, . . . , σr), σi > 0 for any i. Thus, we can write that:

Tk+1 := ϕ(Tk) = Tk (I + (I − DTk)) [(I + χ(I − DTk)2 + (I − DTk)

4)

×(I + κ(I − DTk)2 + (I − DTk)

4)].(27)

Applying mathematical induction, one can write that

(V−1N)X0(M−1(U∗)−1) = λ(V−1N)A#(M−1(U∗)−1)

= λ(V−1N)N−1 A∗(MM−1(U∗)−1)

= λ(V−1N)N−1Vdiag(D, 0)U∗(MM−1(U∗)−1)

= diag(λD, 0).

(28)

390


In addition, when (31) is satisfied, then using (15), one can get that:

(V−1N)Xk+1(M−1(U∗)−1) = (V−1N)Xk(M−1(U∗)−1)

× (2I − (V−1N)AXk(M−1(U∗)−1))

×[(I + χ(V−1N)(I − AXk)2(M−1(U∗)−1)

+(V−1N)(I − AXk)4(M−1(U∗)−1))

×(I + κ(V−1N)(I − AXk)2(M−1(U∗)−1)

+(V−1N)(I − AXk)4)(M−1(U∗)−1)].

(29)

Using the fact that A = U∗MUdiag(D, 0) = V∗NV, one attains

(V−1N)Xk+1(M−1(U∗)−1) = diag(ϕ(Tk), 0), (30)

which shows that (27) is a diagonal matrix. This completes the proof.

3. Error Analysis

The objective of this section is to provide a matrix analysis for the convergence of the iterationscheme (15).

Theorem 1. Let us consider that A is an m× n matrix whose WSVD is provided by (4). Furthermore, assumethat the starting value is given by (11). Thus, the matrix sequence from (15) tends to A†

MN.

Proof. In light of (4), to prove our convergence for the WMP inverse, we now just need to prove that

limk→∞

(V−1N)Xk(M−1(U∗)−1) = diag(D−1, 0). (31)

It is obtained, using Lemmas 1 and 2, that

Tk = diag(τ(k)1 , τ

(k)2 , . . . , τ

(k)r ), (32)

whereτ(0)i = λσi (33)

andτ(k+1)i = τ

(k)i

(2I + σiτ

(k)i

)[(I

+χ(σiτ(k)i )2 + (σiτ

(k)i )4)(I

+κ(σiτ(k)i )2 + (σiτ

(k)i )4)].

(34)

The sequence produced by (34) is the result of employing (15) in calculating the zero σ−1i of

the functionφ(τ) = σi − τ−1, (35)

using the starting condition τ(0)i .

We observe that convergence to σ−1i can be achieved, as long as

0 < τ(0)i <

2σi

, (36)

which results in a criterion on λ (the selection in formula (12) has now been shown). Hence,{Tk} → Σ−1, and (31) is satisfied. It is now clear that {Xk}k=∞

k=0 → A†MN when k → ∞. This concludes

the proof.

391



In this section, our aim is to study the efficiency of the proposed approach for calculating theWMP inverse computationally and analytically. To do this, we considered several competitors fromthe literature in our comparisons, such as those from (6), (7), (10), and (15), denoted by “SM2”, “CM3”,“KMS10”, and “PM10”, respectively.

Note that all computations were done in Mathematica 11.0 [30] and the time is reported in seconds.The hardware used was a CPU Intel Core i5 2430-M with 16 GB of RAM.

We know that the efficiency index is expressed by [31]:

EI = ρ1κ , (37)

where ρ and κ stand for the speed and the whole cost in each cycle, respectively.As such, the efficiency index of different methods (6–10) and (15) are reported by: 2

12 � 1.414,

313 � 1.442, 3

14 � 1.316, 4

14 � 1.414, 10

110 � 1.258, and 10

16 � 1.467, respectively. Clearly, our

investigated iterative expression has better a index and can be more useful in finding the WMP inverse.

Example 1. [29] The purpose of this experiment was to examine the calculation of WMP inverses for 10 uniformrandomly provided m1× n1 = 200× 210 matrices, as follows:

SeedRandom[12]; no = 10; m1 = 200; n1 = 210;

ParallelTable[A[k] = RandomReal[{1}, {m1, n1}];, {k, no}];

where the ten various HPD matrices M and N were given by:

ParallelTable[MM[k] = RandomReal[{2}, {m1, m1}];, {k, no}];

ParallelTable[MM[k] = Transpose[MM[k]].MM[k];, {k, no}];

ParallelTable[NN[k] = RandomReal[{3}, {n1, n1}];, {k, no}];

ParallelTable[NN[k] = Transpose[NN[k]].NN[k];, {k, no}];

The results by applying the stop termination

||Xk+1 − Xk||2 ≤ 10−10, (38)

are reported in Tables 1 and 2, based on the number of iterations, elapsed CPU time (in seconds), andX0 = 1

σ21

A#. As can be observed from the results, the best scheme in terms of number of iterations and

time was (15).

Table 1. Comparison based on the number of iterations and the required mean in Experiment 1.

Methods SM2 CM3 KMS10 PM10

A1 68 43 22 22A2 69 44 22 22A3 67 43 21 21A4 71 46 23 23A5 72 46 23 23A6 72 46 23 23A7 66 42 21 21A8 78 50 25 25A9 63 41 20 20A10 69 44 22 22

Mean 69.5 44.5 22.2 22.2

392


Table 2. Comparison based on the elapsed CPU time and its mean in Experiment 1.


A1 1.4954 1.04826 0.996155 0.755317A2 1.4563 1.08006 0.984057 0.767785A3 1.37301 1.03847 0.967427 0.720294A4 1.53201 1.10927 1.0365 0.789994A5 1.50908 1.10164 0.998853 0.794098A6 1.51215 1.11421 1.03361 0.823177A7 1.39481 1.00116 0.915244 0.743779A8 1.62742 1.24438 1.12434 0.87007A9 1.32916 0.999683 0.903072 0.709523A10 1.49738 1.05736 0.985084 0.764156

Mean 1.47267 1.07945 0.994434 0.773819

Example 2. The iterative methods were compared for five randomly generated dense m1× n1 = 500× 500matrices produced in Mathematica environment by the following piece of code:

m1 = 500; n1 = 500; no = 5; SeedRandom[12];

ParallelTable[A[k] = RandomReal[{0, 1}, {m1, n1}];, {k, no}];

ParallelTable[MM[k] = RandomReal[{0, 1}, {m1, m1}];, {k, no}];

ParallelTable[MM[k] = Transpose[MM[k]].MM[k];, {k, no}];

ParallelTable[NN[k] = RandomReal[{0, 1}, {n1, n1}];, {k, no}];

ParallelTable[NN[k] = Transpose[NN[k]].NN[k];, {k, no}];

Here, we applied the stopping condition

||Xk+1 − Xk||∞ ≤ 10−10, (39)

with a change in the initial approximation as X0 = 1.5σ2

1A#. Noting that the weights M and N were very

ill-conditioned, as we had produced them to be. We report the results in Tables 3 and 4, which revealthat the novel approach was superior to the existing solvers.

Table 3. Comparison based on the number of iterations and the required mean in Experiment 2.


A1 98 61 30 30A2 86 55 27 27A3 83 53 26 26A4 85 54 27 27A5 81 52 26 26

Mean 86.6 55. 27.2 27.2

Table 4. Comparison based on the elapsed time and its mean in Experiment 2.


A1 7.89745 7.20885 12.1801 7.11963A2 6.90346 6.6397 10.933 6.50042A3 2.34013 2.23622 3.75341 2.20977A4 2.23133 2.15679 3.78848 2.23819A5 2.44316 2.26733 3.79153 2.2391

Mean 4.36311 4.10178 6.88929 4.06142

One other application of (15), aside from computing the WMP inverse, is in finding goodapproximate inverse pre-conditioners for Krylov methods when tackling large sparse linear system of

393


equations (see, e.g., [29]). In fact, to apply our scheme in such environments, we can employ severalcommands, such as SparseArray[] for handling sparse matrices.

The main advantage of the proposed method is the improvement of convergence order obtainedby improving the computational efficiency index. Although this computational efficiency indeximprovement was not observed to be drastic, in solving practical problems in higher dimensions itleads to a clear reduction of computation time.

5. Ending Notes

We have investigated a tenth order iterative method for computing the WMP inverse requiringonly six matrix products. The WMP inverse has many applications, from the numerical solution ofnon-linear equations (those involving singular linear systems [32]) to direct engineering applications.Clearly, the efficiency index will reach 101/6 � 1.46, which is better than the Newton–Schulz andChebyshev methods for calculating the WMP inverse. The convergence order of the scheme wassupported and upheld analytically. The extension of this improved version of the hyperpower familyfor computing other types generalized inverses, such as outer and inner inverses, under special criteriaand initial matrices provides a direction for future works in this active topic of research.

Funding: This research project was supported by a grant from the “Research Center of the Female Scientific andMedical Colleges”, Deanship of Scientific Research, King Saud University.


References

1. Bin Jebreen, H.; Chalco-Cano, Y. An improved computationally efficient method for finding the Drazininverse. Disc. Dyn. Nat. Soc. 2018, 2018, 6758302. [CrossRef]

2. Niazi Moghani, Z.; Khanehgir, M.; Mohammadzadeh Karizaki, M. Explicit solution to the operator equationAXD + FX∗B = C over Hilbert C∗-modules. J. Math. Anal. 2019, 10, 52–64.

3. Stanimirovic, P.S.; Katsikis, V.N.; Srivastava, S.; Pappas, D. A class of quadratically convergent iterativemethods. RACSAM 2019, 1–22. [CrossRef]

4. Ben-Israel, A.; Greville, T.N.E. Generalized Inverses: Theory and Applications, 2nd ed.; Springer: New York, NY,USA, 2003.

5. Godunov, S.K.; Antonov, A.G.; Kiriljuk, O.P.; Kostin, V.I. Guaranteed Accuracy in Numerical Linear Algebra;Springer: Dordrecht, The Netherlands, 1993.

6. Fredholm, I. Sur une classe d’équations fonctionnelles. Acta Math. 1903, 27, 365–390. [CrossRef]7. Wang, G.R. A new proof of Grevile’s method for computing the weighted M-P inverse. J. Shangai Norm. Univ.

1985, 3, 32–38.8. Bakhtiari, Z.; Mansour Vaezpour, S. Positive solutions to the system of operator equations TiX = Ui and

TiXVi = Ui. J. Math. Anal. 2016, 7, 102–117.9. Xia, Y.; Chen, T.; Shan, J. A novel iterative method for computing generalized inverse. Neural Comput. 2014,

26, 449–465. [CrossRef]10. Courriee, P. Fast computation of Moore-Penrose inverse matrices. arXiv preprint 2008, arXiv:0804.4809.11. Lu, S.; Wang, X.; Zhang, G.; Zhou, X. Effective algorithms of the Moore-Penrose inverse matrices for extreme

learning machine. Intell. Data Anal. 2015, 19.4, 743–760. [CrossRef]12. Sheng, X.; Chen, G. The generalized weighted Moore-Penrose inverse. J. Appl. Math. Comput. 2007, 25,

407–413. [CrossRef]13. Soleymani, F.; Soheili, A.R. A revisit of stochastic theta method with some improvements. Filomat 2017, 31,

585–596. [CrossRef]14. Söderström, T.; Stewart, G.W. On the numerical properties of an iterative method for computing the

Moore-Penrose generalized inverse. SIAM J. Numer. Anal. 1974, 11, 61–74. [CrossRef]15. Stanimirovic, P.S.; Ciric, M.; Stojanovic, I.; Gerontitis, D. Conditions for existence, representations, and

computation of matrix generalized inverses. Complexity 2017, 2017, 6429725. [CrossRef]

394


16. Gulliksson, M.E.; Wedin, P.A.; Wei, Y. Perturbation identities for regularized Tikhonov inverse and weightedpseudo inverse. BIT 2000, 40, 513–523. [CrossRef]

17. Roy, F.; Gupta, D.K.; Stanimirovic, P.S. An interval extension of SMS method for computing weightedMoore-Penrose inverse. Calcolo 2018, 55, 15. [CrossRef]

18. Van Loan, C.F. Generalizing the singular value decomposition. SIAM J. Numer. Anal. 1976, 13, 76–83.[CrossRef]

19. Zhang, N.; Wei, Y. A note on the perturbation of an outer inverse. Calcolo 2008, 45, 263–273. [CrossRef]20. Ghorbanzadeh, M.; Mahdiani, K.; Soleymani, F.; Lotfi, T. A class of Kung-Traub-type iterative algorithms for

matrix inversion. Int. J. Appl. Comput. Math. 2016, 2, 641–648. [CrossRef]21. Pan, V.Y. Structured Matrices and Polynomials: Unified Superfast Algorithms; BirkhWauser: Boston, MA, USA;

Springer: New York, NY, USA, 2001.22. Schulz, G. Iterative Berechnung der Reziproken matrix. Z. Angew. Math. Mech. 1933, 13, 57–59. [CrossRef]23. Li, H.-B.; Huang, T.-Z.; Zhang, Y.; Liu, X.-P.; Gu, T.-X. Chebyshev-type methods and preconditioning

techniques. Appl. Math. Comput. 2011, 218, 260–270. [CrossRef]24. Krishnamurthy, E.V.; Sen, S.K. Numerical Algorithms—Computations in Science and Engineering; Affiliated

East-West Press: New Delhi, India, 1986.25. Sen, S.K.; Prabhu, S.S. Optimal iterative schemes for computing Moore-Penrose matrix inverse. Int. J.

Syst. Sci. 1976, 8, 748–753. [CrossRef]26. Grevile, T.N.E. Some applications of the pseudo-inverse of matrix. SIAM Rev. 1960, 3, 15–22. [CrossRef]27. Huang, F.; Zhang, X. An improved Newton iteration for the weighted Moore-Penrose inverse. Appl. Math.

Comput. 2006, 174, 1460–1486. [CrossRef]28. Sánchez León, J.G. Mathematica Beyond Mathematics: The Wolfram Language in the Real World; Taylor & Francis

Group: Boca Raton, FL, USA, 2017.29. Zaka Ullah, M.; Soleymani, F.; Al-Fhaid, A.S. An efficient matrix iteration for computing weighted

Moore-Penrose inverse. Appl. Math. Comput. 2014, 226, 441–454. [CrossRef]30. Trott, M. The Mathematica Guide-Book for Numerics; Springer: New York, NY, USA, 2006.31. Ostrowski, A.M. Sur quelques transformations de la serie de LiouvilleNewman. CR Acad. Sci. Paris 1938,

206, 1345–1347.32. Soheili, A.R.; Soleymani, F. Iterative methods for nonlinear systems associated with finite difference approach

in stochastic differential equations. Numer. Algor. 2016, 71, 89–102. [CrossRef]


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mathematics

Article

An Improved Curvature Circle Algorithm forOrthogonal Projection onto a Planar Algebraic Curve

Zhinan Wu 1,† and Xiaowu Li 2,*,†

1 School of Mathematics and Computer Science, Yichun University, Yichun 336000, China; [email protected] College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China* Correspondence: [email protected]; Tel.:+86-187-8613-2431† These authors contributed equally to this work.

Received: 10 August 2019; Accepted: 25 September 2019; Published: 1 October 2019

Abstract: Point orthogonal projection onto planar algebraic curve plays an important role in computergraphics, computer aided design, computer aided geometric design and other fields. For the casewhere the test point p is very far from the planar algebraic curve, we propose an improved curvaturecircle algorithm to find the footpoint. Concretely, the first step is to repeatedly iterate algorithm(the Newton’s steepest gradient descent method) until the iterated point could fall on the planaralgebraic curve. Then seek footpoint by using the algorithm (computing footpoint q) where thecore technology is the curvature circle method. And the next step is to orthogonally project thefootpoint q onto the planar algebraic curve by using the algorithm (the hybrid tangent vertical footalgorithm). Repeatedly run the algorithm (computing footpoint q) and the algorithm (the hybridtangent vertical foot algorithm) until the distance between the current footpoint and the previousfootpoint is near 0. Furthermore, we propose Second Remedial Algorithm based on ComprehensiveAlgorithm B. In particular, its robustness is greatly improved than that of Comprehensive AlgorithmB and it achieves our expected result. Numerical examples demonstrate that Second RemedialAlgorithm could converge accurately and efficiently no matter how far the test point is from the planealgebraic curve and where the initial iteration point is.

Keywords: point projection; intersection; planar algebraic curve; Newton’s iterative method;the improved curvature circle algorithm

1. Introduction

Reconstructing curve/surface is an important work in the field of computer aided geometricdesign, especially in geometric modeling and processing where it is crucial to fit curve/surface inhigh accuracy and reduce the error of representation curve/surface. The representation of the fourcurve types are the explicit-type, implicit-type, parametric-type and subdivision-type. Because implicitrepresentation has unique advantage in the process of computer aided geometric design, it has wideand far-reaching applications. From scattered and unorganized three-dimensional data, Bajaj et al. [1]reconstructed surface and functions on surfaces. They [2,3] have constructed the algebraic B-splinesurfaces with least-squares fitting feature using tensor product technique. Schulz et al. [4] constructedan enveloping algebraic surface using gradually approximate algebraization method. Kanatani et al. [5]applied the algebraic curve to construct geometric ellipse fitting using unified strict maximumlikelihood estimation method. Mullen et al. [6] reconstructed robust and accurate algebraic surfacefunctions to sign the unsigned from scattered and unorganized three-dimensional data point sets.Upreti et al. [7] used a technique to sign algebraic level sets on NURBS surface and algebraic Booleanlevel sets on NURBS surfaces. Rouhani et al. [8] applied the algebraic function for polynomialrepresentation system. And L.G. Zagorchev et al. [9] applied the algebraic function for generalalgebraic surface.



Up to now, there are three main types of methods to solve the problem of point orthogonalprojection onto planar algebraic curve: local method, global method and compromise method betweenthese two methods. Here are three typical approaches.

According to the most basic geometric characteristic, orthogonal projection of test point p ontothe planar algebraic curve is actually the point x on the curve such that cross product of vectors −→xp and∇ f (x) is 0.

∇ f (x)× (p− x) = 0. (1)

Equation (1) can be transformed into Newton’s iteration formula (3). Furthermore, Sullivan et al. [10]adopted a hybrid method with Lagrange multiplier and Newton’s iterative method to compute theclosest point on the planar algebraic curve for each test point. Some orthogonal projection problems canbe transformed into solving system of nonlinear equations. The common characteristic of methods [10,11]is that they converge locally and fast, while methods [10,11] are dependent on the initial points.

The first global method of solving system of nonlinear equations is the Homotopycontinuous method [12,13]. They constructed Homotopy continuous formula.

H(x, t) = (1− t)P(x) + tQ(x), t ∈ [0, 1] (2)

where t is a parameter of continuous transformation from 0 to 1, P(x) = 0 is the original system ofnonlinear equations to be solved, Q(x) is the objective solution of system of nonlinear equationsP(x) = 0. All isolated solutions of system of nonlinear equations P(x) = 0 can be computedby the numerical continuous Homotopy methods [12,13]. So the Homotopy methods [12,13]are global convergence. The Homotopy methods’ robustness is proved by [14] and their hightime-consuming property is verified in [15]. Of course, the Homotopy methods [12,13] are idealin theory, but it is difficult to find or construct the objective system of nonlinear equations Q(x) = 0 inpractical engineering applications.

The second global resultant methods convert system of nonlinear equations into the expressionof the resultants and then solve the resultants [16–19]. According to classical elimination theory,system of two nonlinear equations with two variables can be turned into a resultant polynomial withone variable, which is equivalent to the two simultaneous equations. The Sylvester’s resultant andCayley’s statement of Bézout’s method are the most famous resultant methods [16–19]. Because theresultant methods [16–19] can solve all roots if the degree of the planar algebraic curve is less than 4,they are good global methods. However, if the degree of the planar algebraic curve is more thanquintic, it becomes harder and harder with increasing degree to solve two-polynomial system with theresultant methods.

The third global method is the adoption of the Bézier clipping technique [20–22]. In the first step,solving the nonlinear system of Equation (1) is transformed into solving all roots of Bernstein-Bézierrepresentation with convex hull property. In the second step, if the parts of the domains do notinclude the solution, we clip the parts of the domains by using convex hull box with Bernstein-Bézierform such that the discarded parts of the region has no solution and all the solutions are in theretained parts of the region. In the third step, the de Casteljau subdivision rule is used to segmentthe remaining part of the curve obtained by elimination in step 2. Repeat steps 2 and 3 until wecan find all the solutions to Equation (1). The advantage of this method is that all solutions ofEquation (1) can be found. But this global clipping method has one difficulty: sometimes Equation (1)is difficult or even impossible to convert into Bernstein-Bézier form. For example, specific Equation (1)∇ f (x) × (p− x) = −36p1y17 + 36xy17 + 6p2x5 − 6x5y − 4p1x + 4p2y + 4x2 − 4y2 is impossible toconvert into Bernstein-Bézier form where f (x) = x6 + 4xy + 2y18 − 1 = 0.

The compromise method is between local and global methods. Consisting of the geometricproperty with computing the nearest point is proposed by Hartmann [23,24] named as the first

397


compromise method. Repeatedly run the Newton’s steepest gradient descent method (3) until theiterative point falls on the planar algebraic curve, where the initial iterative point is unrestricted.

xn+1 = xn − ( f (xn)/ 〈∇ f (xn),∇ f (xn)〉)∇ f (xn). (3)

q = p− (〈p− yn,∇ f (yn)〉 / 〈∇ f (yn),∇ f (yn)〉)∇ f (yn). (4)

Running the iterative formula (4) one time, the method [23,24] can obtain the vertical foot pointq where the iterative point yn of the formula (4) is the final iterative point obtained by formula (3).Continuously iterate the above two steps until the vertical foot point q is on the planar algebraiccurve f (x). Unluckily, progressive geometric tangent approximation iteration method with computingvertical foot point q fails for some planar algebraic curves f (x).

The second compromise method is developed by Nicholas [25] who adopted the osculatingcircle technique to realize orthogonal projection onto the planar algebraic curve. Calculate thecorresponding curvature at one point on the planar algebraic curve, and then the radius and center ofthe curvature circle. The line segment formed by the test point and the center of the curvature circleintersects the curvature circle at footpoint q. Approximately take the footpoint q as a point on theplanar algebraic curve. For the new point on the planar algebraic curve, repeat the above procedureto get a new footpoint and corresponding new approximate point on the planar algebraic curve.Repeat the above behavior until the footpoint q is the orthogonal projection point pΓ. Because theplanar algebraic curve does not have parametric control like parametric curve, taking the footpointas an approximate point on the planar algebraic curve will bring about large errors. So it makes theoperation of the whole algorithm unstable.

The third compromise method is the circle shrinking technique [26]. Repeatedly run the iterativeformula (3) such that the final iterative point pc falls on the planar algebraic curve as far as possible,where the selection of initial iterative point is arbitrary. The next iterative point on the planar algebraiccurve is obtained through a series of combined operations of circle and the planar algebraic curve,where the center and radius of the circle are test point p and ‖p− pc‖, respectively. A series ofcombined operations include the two most important steps: Find a point p+ on the circle by means ofthe mean value theorem; Seek the intersection of the line segment pp+ and the circle where we callthis intersection as the current intersection point pc. Repeatedly run this series of combined operationsuntil the distance between the current point pc and the previous point pc is 0. The circle shrinkingtechnique [26] takes a lot of time to seek point p+ each time. The algorithm has one difficulty: if thedegree of the planar algebraic curve is higher than 5, the intersection point pc of line segment pp+ andthe planar algebraic curve cannot be solved directly by formula or the iterative methods to find theintersection pc will lead to instability.

The four compromise method is a circle double-and-bisect algorithm [27]. The circle doublingalgorithm begins with a very small circle where the center is the test point p and the radius is verysmall r1. Keep the same center of the circle, take the radius r2 twice of r1 to draw a new circle. If thereis no intersection between the new circle and the curve, draw a new circle with radius twice of r2.Continuously repeat the above process until new circle can intersect with the planar algebraic curveand the former circle does not. Naturally, the former circle and the current circle are called interiorcircle and exterior circle, respectively. Moreover, the bisecting technology implements the rest ofthe process. Continue to draw a new circle with new radius r = (r1 + r2)/2. If the new current circlewhose radius is r intersects with the curve, substitute r for r2, else for r1. Repeatedly run the aboveprogress until the difference between the two radii is approximate zero(|r1 − r2| < ε). But this methodis very difficult to judge whether the exterior circle intersects the planar algebraic curve or not [27].

The fifth compromise method is the integrated hybrid second order algorithm [28]. It includes twosub-algorithms: the hybrid second order algorithm and the initial iterative value estimation algorithm.They mainly exploint three ideas: (1) the tangent orthogonal vertical foot method coupled withcalibration method; (2) Newton’s steepest gradient descent iterative method to impel the iteration point

398


to be on the planar implicit curve; (3) Newton’s iterative method to speed up the whole iteration process.Before running the hybrid second order algorithm, the initial iterative value estimation algorithm isused to force the initial iterative value of the formula (17) of the hybrid second order algorithm andthe orthogonal projection point pΓ as close as possible. After a lot of tests, if the distance betweenthe test point p and the curve is not very far, the advantages of this algorithm are obvious in term ofrobustness and efficiency. But when the test point is very far from the curve, the integrated hybridsecond order algorithm is invalid.

2. The Improved Curvature Circle Algorithm

In Reference [28], when the test point p is not particularly far away, the integrated hybridalgorithm can have ideal result. But if the test point p is very far from the curve, the algorithm isinvalid where the test point p can not be robustly and effectively orthogonally projected onto theplanar algebraic curve. In order to overcome this difficulty, we propose an improved curvature circlealgorithm to ensure robustness and effective convergence with the test point p being arbitrarily faraway. No matter how far the test point p is from the planar algebraic curve, if the initial iteration pointx0 is very close to the orthogonal projection point of the test point p, the preconceived algorithm canconverge well. So we attempt to construct an algorithm to find an initial iterative point very close tothe orthogonal projection point pΓ of the test point p. The general idea is the following. Repeatedlyiterate the formula (3) by utilizing the Newton’s steepest gradient descent method until the iterationpoint fall on the planar algebraic curve as far as possible, written as pc. This time, the distance betweenthe iteration point pc and the orthogonal projection point pΓ is much smaller than that between theoriginal iteration point x0 and the orthogonal projection point pΓ. The iteration point pc is closer to theorthogonal projection point pΓ. In order to further promote the iteration point pc and the orthogonalprojection point pΓ to be closer, we introduce a key step with curvature circle algorithm. Draw acurvature circle through point pc on the planar algebraic curve with the radius R determined by thecurvature k and the center m being a normal direction point of point pc on the planar algebraic curve.Line segment mp determined by the test point p and the center m intersects curvature circle at pointq. We take the intersection point q as the next iteration point for the iteration point pc. Of course,the distance between the intersection point q and the orthogonal projection point pΓ is much smallerthan the previous one. We use the intersection point q as the new test point, and run the hybridalgorithm again where the initial iterative point at this moment can be set as q− (0.1, 0.1). Repeatedlyiterate until the iteration point falls on the planar algebraic curve f (x), written as pc. We repeat thelast two key steps in this procedure until the iteration point pc and the orthogonal projection point pΓoverlap (See Figure 1).

p

pΓ

f x( )

Figure 1. Test point p orthogonal projection onto planar algebraic curve f (x).

Let’s elaborate on the general idea. Let p be a test point on the plane. There is an planar algebraiccurve Γ on the plane.

f (x, y) = 0. (5)

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The plane algebraic curve (5) can be simply written as

f (x) = 0, (6)

where x = (x, y). The goal of this paper is to find a point pΓ on the planar algebraic curve f (x) tosatisfy the basic relationship

‖p− pΓ‖ = minx∈Γ

‖p− x‖ . (7)

The above problem can be written as⎧⎪⎨⎪⎩f (pΓ) = 0,∇ f (pΓ)× (p− pΓ) = 0,‖p− pΓ‖ = min

x∈Γ‖p− x‖ ,

(8)

where ∇ f =

[∂ f∂x

,∂ f∂y

]is Hamiltonian operator and symbol × is cross product. We take s as the arc

length parameter of the planar algebraic curve f (x) and t =

[dxds

,dyds

]is unit tangent vector along the

planar algebraic curve f (x). Take derivative of Equation (6) with respect to arc length parameter s andcombine with unit tangent vector condition ‖t‖ = 1, we obtain the following simultaneous system ofnonlinear equations, {

〈t,∇ f 〉 = 0,‖t‖ = 1.

(9)

It is easy to get the solution of Equation (9).

t =

[−∂ f

∂y,

∂ f∂x

]/ ‖∇ f ‖ . (10)

Repeatedly iterate Equation (3) called as the Newton’s steepest gradient descent method until untilthe iterative termination criteria | f (xn+1)| < ε, where the initial iterative point is x0 = p− (0.1, 0.1)and refer to the iterative point xn+1 as pc. The first advantage of the Newton’s steepest gradient descentmethod (3) is to make the iteration point fall on the planar algebraic curve f (x) as far as possible. Itssecond advantage of the Newton’s steepest gradient descent method (3) is that the iteration pointfallen on the planar algebraic curve is relatively close to the orthogonal projection point pΓ, and itbrings great convenience to implementation of the subsequent sub-algorithms. The Newton’s steepestgradient descent method (Algorithm 1) can be specifically described as (See Figure 2).

Algorithm 1: The Newton’s steepest gradient descent method.Input: The test point p and the planar algebraic curve f (x) = 0Output: The iterative point pc fallen on planar algebraic curve f (x) = 0Description:

Step 1:

xn+1 = p− (0.1, 0.1);Do {

xn = xn+1 ;Update xn+1 according to the iterative Equation (3);

}while (| f (xn+1)| > ε&& ‖xn+1 − xn‖ > ε);Step 2:

pc = xn+1 ;Return pc;

400


x0

x1

x2

xn + 1

(a)

p

pΓf x( )

pc

(b)

Figure 2. The entire graphic demonstration of Algorithm 1. (a) The whole iterative process of theNewton’s steepest gradient descent method; (b) The last step of the iterative point pc fallen on theplanar algebraic curve f (x) through the Newton’s steepest gradient descent method.

In this case, if the iterative point pc fallen on the planar algebraic curve f (x) is taken as the initialiterative point of the hybrid algorithm, convergence or divergence may occur where divergence can notimprove the algorithm. As for divergence, it can not achieve the purpose of improving the algorithm.From another point of view, the distance between iteration point pc and orthogonal projection pointpΓ of the test point p should be closer. It lays a good foundation for the implementation of subsequentsub-algorithms. In order to get the iteration point and the orthogonal projection point pΓ closer,we adopt curvature circle way to promote the iteration point and the orthogonal projection point pΓbeing closer. Because the iterative point is on the planar algebraic curve, the curvature k at the iterativepoint pc fallen on the planar algebraic curve f (x) is defined as [29],

k = k(x, y) =

[− fy, fx]

G[− fy, fx

]T

‖∇ f ‖3 , (11)

where G =

(fxx fxy

fyx fyy

). The radius R and the center m of the curvature circle © directed by the

curvature k areR = |1/k| , (12)

and

m = pc +−→nk

, (13)

where the unit normal vector −→n is −→n =∇ f‖∇ f ‖ . The line segment mp determined by the test point p

and the center m of the curvature circle© intersects the curvature circle© at point q which is namedas footpoint q. From elementary geometric knowledge, the parametric equation of the line segmentmp can be expressed as

x = p + (m− p)w, (14)

where parametric 0 ≤ w ≤ 1 is undetermined. In addition, the equation of the curvature circle© canbe written as

‖m− x‖ = R. (15)

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By solving Equation (14) and Equation (15) together, the analytic expression of the intersection q

is obtainedq = p + (m− p)w, (16)

where the undetermined parameter w is accurately identified as w = 1− R‖m− p‖ . The computation

of the footpoint q can be realized through Algorithm 2 (See Figure 3).

Algorithm 2: Computing footpoint q via the curvature circle© and the line segment mp.Input: The test point p, the planar algebraic curve f (x) = 0 and the iterative point pc on theplanar algebraic curve f (x) = 0.

Output: The footpoint q.Description:

Step 1:

Compute the curvature k of the iterative point pc fallen on the planar algebraic curvef (x) = 0 by the curvature calculation formula (11).

Step 2:

Calculate the radius R and the center m of the curvature circle© through the formulas(12) and (13), respectively.

Step 3:

Compute the footpoint q by the formula (16).Return q;

Remark 1. The important formula for computing the curvature k is the formula (11). If the denominator ofthe curvature k with the formula (11) is 0, the whole iteration process will degenerate. In order to solve thisspecial degeneration, we adopt a small perturbation of the curvature k of the formula (11) in programmingimplementation of Algorithm 2. Namely, the denominator of the curvature k with the formula (11) could beincremented by a small positive constant ε, the denominator of the curvature k is the denominator of the curvaturek +ε, and Algorithm 2 continues to calculate the center and the radius of the curvature circle correspondingto the curvature after disturbance. Of course, in all subsequent formulas or iterative formulas, we also dothe same denominators perturbation treatment for the case of the zero denominators of the formulas or theiterative formulas.

p

pΓ

f x( )

pc

m

q

Figure 3. Graphic demonstration for Algorithm 2.

Under this circumstance, if the footpoint point q at this moment is taken as the initial iterationpoint of the hybrid algorithm, the convergence probability of the hybrid algorithm is much greater than

402


that of using the point pc in Algorithm 1 as the initial iterative point of the hybrid algorithm. The reasonis that the distance ‖q− pΓ‖ is smaller than the distance ‖pc − pΓ‖. But divergence may happen inthis case. In order to further guarantee the robustness,we orthogonally project the footpoint q onto theplanar algebraic curve f (x) by using the hybrid algorithm, instead of directly using the footpoint q

as the initial iterative point. At this time we still call the orthogonal projection point of the footpointq as the point pc which is just fallen on the planar algebraic curve f (x). Because at this time thefootpoint q is close to the planar algebraic curve f (x), the algorithm can ensure complete convergence.The distance between the iterative point pc and the orthogonal projection point pΓ of the test point p

becomes smaller again. The core iterative formula (17) of the hybrid algorithm is as follows (See [28]).⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

yn = xn − ( f (xn)/ 〈∇ f (xn),∇ f (xn)〉)∇ f (xn),zn = yn − (F(yn)/ 〈∇F(yn),∇F(yn)〉)∇F(yn),Q = q− (〈(q− zn),∇ f (zn)〉 / 〈∇ f (zn),∇ f (z n)〉)∇ f (zn),un = zn+sign(〈q− zn, t0〉)t0Δs,vn = un − (F(un)/ 〈∇F(un),∇F(un)〉)∇F(un),

xn+1= vn + [−Δe, 0][∇ f T , (Δvn)

T]−1

(i f∣∣[∇ f T , (Δvn)T]∣∣ = 0, xn+1 = vn),

(17)

where F0(x) = [(q− x)×∇ f (x)] = 0, F(x) =F0(x)√

〈∇ f (x),∇ f (x)〉, t0 =

[− fy, fx]

‖Δ f ‖ , Δs = ‖Q− zn‖,

f (vn) = Δe, Δvn = −(F(un)/ 〈∇F(un),∇F(un)〉)∇F(un).The iterative formula (17) mainly contains four techniques. The core technology is the tangent

foot vertical method with the third step and the fourth step of the iterative formula (17). Draw atangent line L from a point on a plane algebraic curve f (x). Through the footpoint q (The footpoint q

at this time is as the test point of iterative formula (17)), make a vertical line of the tangent L and get itscorresponding vertical foot point Q, which is equivalent to the third step in the formula (17). From thefourth step of the iterative formula (17), we get the next iteration point of particular importance for theinitial iteration point. When the next iteration point is not very close to the planar algebraic curve f (x),we adopt the second important technique with the iteration point correction method, equivalent tothe sixth step of the iterative formula (17). The iteration point is to move to the plane algebra curveas close as possible such that the distance between the correction point of the iteration point and theplane algebra curve f (x) is as close as possible. These two techniques are pure geometric techniques.When the distance between the test point and the planar algebraic curve is very close, the effect ofconvergence is obvious. Of course, when the distance between the test point and the planar algebraiccurve is relatively long, sometimes there will be non-convergence. In order to improve the robustnessof convergence, we add the Newton’s steepest gradient descent method before the first technique withthe third step and the fourth step of the iterative formula (17). Its first aim is to bring the initial iterationpoint closer to the planar algebraic curve f (x). Its second aim is to promote the accuracy of subsequentiterations. In order to accelerate the whole iteration process of the iterative formula (17), we onceagain incorporate the fourth technology of Newton’s iterative method which is closely related to thefootpoint q. This technique not only accelerates the convergence rate of the whole iteration process butalso improves the iteration robustness. Furthermore, the accuracy of the whole iteration process can beimproved by the fourth technique. So we add Newton’s iterative method after the first step with thesecond technique, and then add it again before the last step with the third technique. Based on theabove explanation and illustration, we get the following the hybrid tangent vertical foot algorithm(Algorithm 3).

403


Algorithm 3: The hybrid tangent vertical foot algorithm (See Figure 4).Input: The footpoint q and the planar algebraic curve f (x) = 0.Output: The point pc fallen on the planar algebraic curve f (x) = 0.Description:

Step 1:

xn+1 = q− (0.1, 0.1);Do {

xn = xn+1;Execute xn+1 according to the iterative Equation (17);

}while (‖xn+1 − xn‖2 > ε&&| f (xn+1)| > ε)

Step 2:

pc = xn+1;Return pc;

With the description of the above three algorithms, we propose a comprehensive and completealgorithm (Algorithm 4) closely related to Algorithm 2 (See Figure 4).

Algorithm 4: The first improved curvature circle algorithm (Comprehensive Algorithm A).Input: Test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p which orthogonally projects the testpoint p onto a planar algebraic curve f (x).

Description:

Step 1: Starting from the neighbor point of the test point p, calculate the point pc fallen on thef (x) via Algorithm 1.

Do{Step 2: Compute the footpoint q via Algorithm 2.Step 3: Project footpoint q onto the planar algebraic curve f (x) via Algorithm 3, then get

the new iterative point pc fallen on the f (x).}while (distance (the current pc, the previous pc)> ε).pΓ = pc;Return pΓ;

Through a series of rigorous deductions, Comprehensive Algorithm A is the important algorithmof our paper. No matter how far the test point p is from the planar algebraic curve f (x), test pointp could very robustly orthogonally projects onto the planar algebraic curve f (x). This has achievedour desired result. After a lot of testing and observation, when the point on the curve is close to theorthogonal projection point, we find that Comprehensive Algorithm A presents two characteristics:(1) difference between the first distance and the second distance decreases slower and slower, where thefirst distance and the second distance are the one between the previous iterative point pc on the planaralgebraic curve and the orthogonal projection point pΓ, and the one between the current iterative pointpc on the planar algebraic curve and the orthogonal projection point pΓ, respectively; (2) the rate goeseven slower at which the absolute value of the inner product gradually approaches zero. These twocharacteristics are what we don’t want to obtain because they are contrary to the efficiency of computersystems. On the premise of ensuring robustness, we try our best to improve and excavate a certaindegree of efficiency for the problem of point orthogonal projection onto planar algebraic curve.We have an ingenious discovery. After each running of Algorithm 3, we run the Newton’s iterativemethod associated with the original test point p, which can improve the convergence and ensurethe orthogonality. Namely, that is to add this step after the last step of the formula (17). Thus theiterative formula (18) is obtained.

404


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

yn = xn − ( f (xn)/ 〈∇ f (xn),∇ f (xn)〉)∇ f (xn),zn = yn − (F(yn)/ 〈∇F(yn),∇F(yn)〉)∇F(yn),Q = q− (〈(q− zn),∇ f (zn)〉 / 〈∇ f (zn),∇ f (z n)〉)∇ f (zn),un = zn+sign(〈q− zn, t0〉)t0Δs,vn = un − (F(un)/ 〈∇F(un),∇F(un)〉)∇F(un),

wn= vn + [−Δe, 0][∇ f T , (Δvn)

T]−1

(i f∣∣[∇ f T , (Δvn)T]∣∣ = 0, then wn = vn),

xn+1 = wn − (G(wn)/ 〈∇G(wn),∇G(wn)〉)∇G(wn),

(18)

where F0(x) = [(q− x)×∇ f (x)] = 0, F(x) =F0(x)√

〈∇ f (x),∇ f (x)〉, t0 =

[− fy, fx]

‖Δ f ‖ , Δs = ‖Q− zn‖,

f (vn) = Δe, Δvn = −(F(un)/ 〈∇F(un),∇F(un)〉)∇F(un), G0(x) = [(p− x)×∇ f (x)] = 0, G(x) =G0(x)√

〈∇ f (x),∇ f (x)〉. Because the iterative formula (17) of Algorithm 3 naturally becomes the iterative

formula (18), so Algorithm 3 naturally becomes the following Algorithm 5.

Algorithm 5: The hybrid tangent vertical foot algorithm.Input: The footpoint q and the planar algebraic curve f (x) = 0.Output: The point pc fallen on planar algebraic curve f (x) = 0.Description:

Step 1:

xn+1 = q− (0.1, 0.1);Do {

xn = xn+1;Execute xn+1 according to the iterative Equation (18);

}while (‖xn+1 − xn‖2 > ε&&| f (xn+1)| > ε)

Step 2:

pc = xn+1;Return pc;

Now let’s replace Algorithm 3 of Comprehensive Algorithms A with Algorithm 5. We get thefollowing Comprehensive Algorithm B (Algorithm 6).

Algorithm 6: The second improved curvature circle algorithm (Comprehensive Algorithm B).Input: Test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p which orthogonally projects the testpoint p onto the planar algebraic curve f (x).

Description:

Step 1: Starting from the neighbor point of the test point p, calculate the point pc fallen on thef (x) via Algorithm 1.

Do{Step 2: Compute the footpoint q via Algorithm 2.Step 3: Project footpoint q onto the planar algebraic curve f (x) via Algorithm 5, then get

new point pc fallen on the f (x).}while(distance(the current pc, the previous pc)> ε).pΓ = pc;Return pΓ;

405


q

yn

(a)

yn

zn

(b)

p

pΓ

f x( )

pc

q

zn

Qz

’

n

(c)

p

pΓ

f x( )

pc

q

zn

Qz

’

nun

(d) (e)

p

pΓ

f x( )

pc

q

zn

Qz

’

nun

vn

xn+1

(f)

Figure 4. Graphic interpretation of the whole iteration process in Algorithm 3. (a) Newton’s steepestgradient descent method in the first step; (b) The Newton’s iteration method related to the test point inthe second step; (c) The vertical foot point Q being the footpoint q orthogonal projection onto tangentline induced by the iterative point zn on the planar algebraic curve in the third step; (d) Calculating lineincremental iterative value in the fourth step; (e) Once again running the Newton’s iteration methodrelated to the test point in the fifth step; (f) Correcting the previous iteration value to improve therobustness of iteration in the last step.

Comprehensive Algorithm A and Comprehensive Algorithm B share common advantage:the robustness of the two algorithms is substantially improved than that of the existing algorithmsbecause our algorithms are not subject to any restrictions on test points and initial iteration points.By comparison, Comprehensive Algorithm B has four advantages over Comprehensive Algorithm A.(1) The last step of the iterative formula (18) in Comprehensive Algorithm B can make correctionscontinuously; (2) The last step of the iterative formula (18) in Comprehensive Algorithm B acceleratesthe whole Comprehensive Algorithm B; (3) The last step of the iterative formula (18) in ComprehensiveAlgorithm B accelerates the inner product of two vectors to 0, where the first vector refers to thevector connecting the test point p and the iteration point zn+1 of Comprehensive Algorithm B andthe second vector

[− ∂ f

∂y , ∂ f∂x

]|x=xn+1

is the tangent vector derived from the iteration point xn+1 on the

planar algebraic curve, respectively; (4) Comprehensive Algorithm B overcomes two shortcomings ofComprehensive Algorithm A.

Of course, when the test point is not too far from the plane algebra curve, ComprehensiveAlgorithm is also convergent for any initial iterative point. However, Comprehensive Algorithm Atakes more time than directly using the hybrid second order algorithm. In practical applications suchas computer graphics, it’s hard to know if the test point p is close to or far from a planar algebraiccurve. Because the main reason is that the degree and the type of the planar algebraic curve restrict therelative distance between the test point p and the planar algebraic curve. In order to optimize timeefficiency, we take advantage of Comprehensive Algorithm A and the hybrid second order algorithmsuch that no matter where the test point p is located, it can be orthogonally projected onto the planaralgebraic curve efficiently and robustly. First, the hybrid second order algorithm is iterated. If it doesnot converge after 100 iterations, it will be changed to Comprehensive Algorithm A to iterate until the

406


iteration point reaches the orthogonal projection point pΓ. Specific algorithm implementation is thefollowing Comprehensive Integrated Algorithm A (Algorithm 7).

Algorithm 7: The first comprehensive integrated improved curvature circle algorithm(Comprehensive Integrated Algorithm A).

Input: Test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p.Description:

Step 1:

xn+1 = p− (0.1, 0.1);for(i = 0; i < N; i ++) {

xn = xn+1;

xn+1=Hybrid second order algorithm( f , p, xn);if(‖xn+1 − xn‖ < ε) break ;

}Step 2:

if(i ≥ N&&d ≥ 1e− 15) {xn = xn+1 ;

xn+1=Comprehensive Algorithm A( f , p, xn);}pΓ = xn+1;Return pΓ;

Number N is an empirical value of the iterative times where the value N is specified as 5 or 6.Similar to Comprehensive Algorithm A, by replacing Algorithm 3 with Algorithm 5, the following

Comprehensive Integrated Algorithm B (Algorithm 8) can be obtained naturally.

Algorithm 8: The second comprehensive integrated improved curvature circle algorithm(Comprehensive Integrated Algorithm B).

Input: The test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p.Description:

Step 1:

xn+1 = p− (0.1, 0.1);for(i = 0; i < N; i ++) {

xn = xn+1;

xn+1=Hybrid second order algorithm( f , p, xn);if(‖xn+1 − xn‖ < ε) break ;

}Step 2:

if(i ≥ N&&d ≥ 1e− 15) {xn = xn+1 ;

xn+1=Comprehensive Algorithm B( f , p, xn);}pΓ = xn+1;Return pΓ;

Number N is an empirical value of the iterative times where the value N is specified as 5 or 6.

407


To sum up, we have presented four synthesis algorithms altogether. After analysis and judgment,Comprehensive Algorithm B and Comprehensive Integrated Algorithm B are the most robustand efficient. On the problem of orthogonal projection of point onto planar algebraic curve, if thedistance between the test point and the planar algebraic curve is close, we recommend the hybridsecond order algorithm, if the distance between the test point and the planar algebraic curve is notclose, we recommend Comprehensive Algorithm B. Of course, if the distance between the test pointand the planar algebraic curve cannot be known to be very far or close, Comprehensive IntegratedAlgorithm B is the best choice.

Remark 2. In sum, Comprehensive Algorithm B has strong superiority over existing algorithms [10–28].If the distance between the test point and the planar algebraic curve is very far away, the test point canbe ideally orthogonally projected onto the planar algebraic curve. But when there are singular points∂ f∂x· ∂ f

∂x+

∂ f∂y· ∂ f

∂y= 0 in the planar algebraic curve, this case will seriously hinder the correct execution

and implementation of Comprehensive Algorithm B. In order to solve the problem in the case of singularities inthe planar algebraic curves, we propose a remedy to Comprehensive Algorithm B (Algorithm 9). The specificdescription is as follows (See Figure 5).

Algorithm 9: The first remedial algorithm of Comprehensive Algorithm B.Input: Test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p.Description:

Step 1.

Starting from the neighbor point of the test point p, calculate the iterative point pc fallenon the planar algebraic curve f (x) via Algorithm 1.

Step 2.

Judge whether to use curvature circle method or tangent method in the next step.Step 3.

Find the left endpoint L0 on the other side of f (x) relative to the test point p. Accordingto the result of step 2, if use curvature circle method, then the left endpoint L0 is equal to theintersection point q which is computed by the curvature circle method with the formula (16).If not, then the left endpoint L0 is equal to the vertical foot Q which is computed by thetangent method with the third step of the formula (17).

Step 4.

Calculate the intersection point pc of the line segment L0p connecting the current leftendpoint L0 and the test point p and the planar algebraic curve f (x) by the hybrid method ofcombining Newton’s iterative method and binary search method. The intersection point pc iscalled as the current iterative point pc;

Step 5.Repeat Step 2,Step 3 and Step 4 until the distance between the current iterative point pc

and the previous iterative point pc is near zero;Step 6.

pΓ= pc;Return pΓ;

Now let’s describe the hybrid method of combining Newton’s iterative method and binary searchmethod in detail. The parameter equation of the line segment L0p can be expressed as{

x = L1 + (p1 − L1)w,y = L2 + (p2 − L2)w,

(19)

408


where L0 = (L1, L2), p = (p1, p2), and 0 ≤ w ≤ 1 is a parameter of Equation (19). SubstituteEquation (19) into Equation (6) of the planar algebraic curve to get a equation on the parameter w,

K(w) = f (x, y) = 0, (20)

where the x and y of Equation (20) are completely determined by the x and y of Equation (19). So themost basic Newton’s iterative formula corresponding to Equation (20) is not difficult to write as,

wn+1 = wn − K(wn)

DK(wn), (21)

where DK(w) is the first derivative of K(w) about the parameter w. Now we start to iterate theNewton’s iterative formula (21) with the initial iterative value w0 = 0.0. Based on the actual situation,the intersection of the line segment L0p and the planar algebraic curve is much closer to the leftendpoint L0 and much farther from the original test point p, therefore, the initial interval of the binarysearch method can be specified as [a, b] = [0.0, 0.5]. The detailed description of the hybrid method ofcombining Newton’s iterative method and binary search method is as following Algorithm 10.

Algorithm 10: The hybrid method of combining Newton’s iterative method and binarysearch method.

Input: The planar algebraic curve f (x), the original test point p = (p1, p2), the iterative pointpc via Algorithm 1.

Output: The intersection pc between the line segment L0p and the planar algebraic curve f (x).Description:

Step 1:

The initial interval of the binary search method [a, b] = [0.0, 0.5], the initial iterativevalue w = 0.0;Step 2:

w = w− K (w) /DK (w);kmin=min(K(a),K(b));kmax=max(K(a),K(b));if (K(w) < kmin or K(w) > kmax)

w = (a + b) /2;sa=sign(K(a));sw=sign(K(w));if(sa == sw)

a = w;else

b = w;Step 3:

Repeatedly iterate Step 2 until |a− b| < ε;Step 4:

pc = L0 + (p− L0)w;Return pc;

409


f(x)

p

m

pc

p’

c

q

pΓ

Q

(a)

pΓ

f( )xm

q

pc LQ

p,

c

p

(b)

Figure 5. Graphical interpretation for the first remedial algorithm of Comprehensive Algorithm B.(a) The intersection point q between the line segment mp and the curvature circle and the test point p

on the opposite side of the planar algebraic curve f (x); (b)The vertical foot Q of the tangential line L

and the test point p on the opposite side of the planar algebraic curve f (x).

The robustness of the first remedial algorithm of Comprehensive Algorithm B is much better thanthat of Comprehensive Algorithm B while the first remedial algorithm of Comprehensive Algorithm Btakes much more time than Comprehensive Algorithm B. The hybrid method of combining Newton’siterative method and binary search method is a hybrid method which binary search method ensures globalconvergence and the Newton’s iterative method plays an accelerating role. In order to ensure robustnessand improve efficiency, we have fully excavated Comprehensive Algorithm B. We have developed SecondRemedial Algorithm (Algorithm 11). The specific description is as follows (See Figure 6).

pΓ

f( )x m

q

pc L

p

pc

Figure 6. Graphic demonstration for Second Remedial Algorithm.

Algorithm 11: Second Remedial Algorithm.Input: Test point p and the planar algebraic curve f (x).Output: Orthogonal projection point pΓ of the test point p which orthogonally projects the testpoint p onto the planar algebraic curve f (x)

Description:

Step 1: Starting from a certain percentage of the test point p, calculate the point pc fallen on thef (x) via Algorithm 1.

Do{Step 2: Compute the footpoint q via Algorithm 2.Step 3: Starting from the footpoint q, compute the iterative point pc fallen on the f (x) via

Algorithm 1.}while(distance(the current pc, the previous pc)> ε).Step 4: Compute the orthogonal projection point pΓ of the test point p via Algorithm 12.

Return pΓ;

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Algorithm 12: The hybrid Newton-type iterative algorithm.Input: The current iterative point pc fallen on the planar algebraic curve f (x) and the planaralgebraic curve f (x).

Output: Orthogonal projection point pΓ of the test point p which orthogonally projects the testpoint p onto the planar algebraic curve f (x).

Description:

Step 1:

xn+1 = pc;Do {

xn = xn+1;Compute xn+1 according to the iterative formula (22);

}while (‖xn+1 − xn‖2 > ε&&| f (xn+1)| > ε)

Step 2:

pΓ = xn+1;Return pΓ;

The expression of the iterative formula (22) is as follow,⎧⎪⎪⎨⎪⎪⎩yn = xn − ( f (xn)/ 〈∇ f (xn),∇ f (xn)〉)∇ f (xn),zn = yn − (F(yn)/ 〈∇F(yn),∇F(yn)〉)∇F(yn),

xn+1 = zn + [−Δe, 0][∇ f T , (Δzn)

T]−1

(i f∣∣[∇ f T , (Δzn)T]∣∣ = 0, then xn+1 = zn),

(22)

where F0(x) = [(p− x)×∇ f (x)] = 0, F(x) =F0(x)√

〈∇ f (x),∇ f (x)〉, f (zn) = Δe, Δzn =

−(F(yn)/ 〈∇F(yn),∇F(yn)〉)∇F(yn).

Remark 3. In this remark, we present the geometric interpretation of Second Remedial Algorithm. The purposeof the first step is to make the iterative point pc fall on the planar algebraic curve as much as possible throughNewton’s steepest gradient descent method of Algorithm 1, where the coordinates of the initial iterativepoint take proportional value of that of the test point p to ensure that Algorithm 1 converges successfully.Otherwise, the distance between the initial iterative point and the planar algebraic curve is very large, whicheasily leads to the divergence of Algorithm 1. The purpose of Do . . . While cycle body in Second RemedialAlgorithm is to continuously and gradually move the iterative point pc to fall on the planar algebraic curve tothe orthogonal projection point pΓ. The second step in Do. . . While cycle body in Second Remedial Algorithmhas two characteristics. Since the footpoint q is the unique intersection point of the curvature circle and thestraight line segment mp connecting the centre m of the curvature circle and the test point q, the footpointq is always closely related to the iterative point pc fallen on the planar algebraic curve and the test point p.The first characteristic is that the footpoint q can guarantee the global convergence of the whole algorithm(Second Remedial Algorithm). The second characteristic is that the distance between the footpoint q and theplanar algebraic curve is much smaller than the distance between the test point p and the planar algebraic curve.So the third step with Algorithm 1 in Do . . . While cycle body can very robustly iterate the footpoint q ontothe planar algebraic curve. The core thought of Do . . . While cycle body in Second Remedial Algorithm is tokeep the iterative point pc to fall on the planar algebraic curve and to move towards the orthogonal projectionpoint pΓ such that the distance ‖pc − pΓ‖ between the iterative point pc and the orthogonal projection pointpΓ becomes smaller and smaller. As the distance ‖pc − pΓ‖ gets smaller and smaller, we have found that thereis a defect in Do . . . While cycle body in Second Remedial Algorithm. The decreasing speed of the distance‖pc − pΓ‖ is getting slower and slower. Especially the second formula of the formula (8) is very difficultto be satisfied. Namely, it is difficult to orthogonalize the vector −→ppc and the vector ∇ f (pc). In order toovercome the difficulty, we add Algorithm 12 behind Do . . . While cycle body in Second Remedial Algorithm.

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Algorithm 12 includes three components: (1) The Newton’s steepest gradient descent method in the first step;(2) The Newton’s iterative method closely associated with the test point p in the second step; (3) Correctingmethod in the third step. Algorithm 12 has four advantages and important roles: (1) Algorithm 12 plays a rolefor accelerating the whole algorithm (Second Remedial Algorithm); (2) The first step plays a role for making theiteration point fall on the planar algebraic curve as far as possible; (3) The second step plays a role for acceleratingorthogonalization between the vector −→ppc and the vector ∇ f (pc); (4) The third step plays a dual role for theaccelerating orthogonalization and the promoting the iterative point to fall on the planar algebraic curve. Thenumerical tests of Second Remedial Algorithm achieve our expected results. No matter how far the test point p isfrom the planar algebraic curve f (x), Second Remedial Algorithm can converge very robustly and efficiently.Second Remedial Algorithm is the best one in our paper. Of course, the robustness and the efficiency of SecondRemedial Algorithm are better than that of the existing algorithms. We are very happy about that.

Remark 4. In order to further improve the efficiency of the test point p orthogonal projecting onto plane algebraiccurve f (x), we construct a Comprehensive Integrated Algorithm C which includes two parts: the hybrid secondorder algorithm in [28] and Second Remedial Algorithm. Firstly run the hybrid second order algorithm in [28].If the hybrid second order algorithm converges, then it means that Comprehensive Integrated Algorithm Cis finished. Otherwise, Second Remedial Algorithm runs until it converges successfully. That is the end ofComprehensive Integrated Algorithm C. The specific description of Comprehensive Integrated Algorithm Cis similar to that of Comprehensive Integrated Algorithm B. Here, we are not giving a detailed description ofComprehensive Integrated Algorithm C. When the distance between the test point p and the planar algebraiccurve f (x) is not far, the hybrid second order algorithm in [28] has very high robustness and efficiency.When the distance between the test point p and the planar algebraic curve f (x) is particularly far, the hybridsecond order algorithm does not converge and fails, while Second Remedial Algorithm converges particularlyrobustly and successfully. To sum up, Comprehensive Integrated Algorithm C absorbs the advantages of twosub-algorithms and overcomes their respective shortcomings such that the robustness and the efficiency ofComprehensive Integrated Algorithm C are maximized.


This section proves that several Comprehensive Algorithms do not depend on the initialiteration points.

Theorem 1. Comprehensive Algorithm A is independent of the initial iterative point.

Proof. Firstly, we state the whole operation process of Comprehensive Algorithm A. ComprehensiveAlgorithm A contains three sub-algorithms (Algorithms 1–3). The role of Algorithm 1 is to repeatedlyiterate the iterative formula (3) through Newton’s steepest gradient descent method such that thefinal iteration point xn+1 could fall on the planar algebraic curve where the final iteration point xn+1

is denoted as pc. The function of Algorithm 2 is to seek the footpoint q. The curvature circle ©determined by the point pc is obtained from the iterative point pc on the planar algebraic curve f (x)of Algorithm 1, where the curvature k, the radius R and the center m are determined by formulas(11)–(13), respectively. The intersection of the line segment mp connecting the center m and the testpoint p and the curvature circle © is foot point q. The footpoint q could be orthogonally projectedonto the planar algebraic curve f (x) by repeated iteration of Algorithm 3 where at this moment thetest point is not the original test point p but the footpoint point q solved by Algorithm 2. Repeatedlyrun Algorithm 2 and Algorithm 3 bound together until the distance between the current footpoint q

and the previous footpoint q is near zero.Secondly, the Comprehensive Algorithm A is independent of the initial iterative point. No matter

how far the original test point p is from the planar algebraic curve f (x), no matter where the initialiterative point x0 is located, Algorithms 1 can ensure that the final iterative point xn+1 or pc of the initialiterative point can fall on the planar algebraic curve f (x). It is obvious that the distance ‖pc − pΓ‖

412


between the iteration point pc and the orthogonal projection point pΓ is much smaller than the distance‖p− pΓ‖ between the orthogonal projection point pΓ and the original test point p. From the iterativepoint pc fallen on the planar algebraic curve f (x), we can calculate the corresponding curvature k andits center m and radius R. The intersection point q of the curvature circle© and the line segment mp

connecting the original test point p and the center m of the curvature circle© is just the footpoint q.That is to say, the footpoint q is directly generated by the curvature circle © and the line segmentmp, and the curvature circle © is controlled by the iterative point pc fallen on the planar algebraiccurve f (x). So the footpoint q is directly controlled by the original test point p and the iterativepoint pc, while the current footpoint q is between the orthogonal projection point pΓ and the currentiterative point pc. It also shows that Algorithm 2 plays a decisive role in the convergence robustness ofComprehensive Algorithm. In addition, we can also know that the distance between the footpointpoint q and the planar algebraic curve f (x) is much smaller than the distance between the originaltest point p and the planar algebra curve f (x). At this point, we keep running Algorithm 3 with thefootpoint point q as the current test point until the current test point can be orthogonally projectedonto the plane algebraic curve f (x) with guaranteed convergence of Algorithm 3. And now we cancall the orthogonal projection point of the footpoint point q as also the iterative point pc fallen onthe planar algebraic curve f (x). The first reason is the distance between the current iterative pointpc and the orthogonal projection point pΓ of the original test point point p is smaller than the onebetween the previous iterative point pc and the orthogonal projection point pΓ of the original testpoint p. The second reason is that it establishes a solid foundation for the convergence robustness of thesubsequent sub-algorithms implementation. Then according to the requirements of ComprehensiveAlgorithm A, the second step and third step of Comprehensive Algorithm A are executed once percycle, the distance ‖pc − pΓ‖ between the current iterative point pc on the planar algebraic curve andthe orthogonal projection point pΓ of the original test point p of the execution result is smaller thanthat between the previous iterative point pc on the planar algebraic curve f (x) and the orthogonalprojection point pΓ of the original test point p. The distance ‖pc − pΓ‖ between the current iterativepoint pc and the orthogonal projection point pΓ of the original test point p is becoming smaller. Sorepeatedly iterate the second step and the third step of Comprehensive Algorithm A until the distance‖pc − pΓ‖ between the current iterative point pc and the orthogonal projection point pΓ of the originaltest point p is becoming smaller and smaller. Ultimately, the distance ‖pc − pΓ‖ between the currentiterative point pc and the orthogonal projection point pΓ of the original test point p is becoming zero. Italso demonstrates that Comprehensive Algorithm A is completely convergent. This further proves thatComprehensive Algorithm A can completely converge no matter how far away the original test point p

is from the planar algebraic curve and no matter where the initial iterative point x0 of ComprehensiveAlgorithm A is on the plane. This means Comprehensive Algorithm A is independent of the initialiterative point.

Theorem 2. Comprehensive Algorithm B is independent of the initial iterative point.

Proof. In the last step of the iterative formula (18) in Algorithm 5, Newton’s iteration method, which isclosely related to the original test point p, is added. In this way, the iterative formula (17) is transformedinto the iterative formula (18) in Algorithm 5. Algorithm 5 has several advantages over Algorithm 3.It can speed up the iteration, improve its accuracy and promote the orthogonalization of the tangentvector derived from the iteration point on the planar algebraic curve and the tangent vector connectingthe test point and the iterative point. Replace Algorithm 3 of Comprehensive Algorithm A withAlgorithm 5 to get Comprehensive Algorithm B. Since Comprehensive Algorithm A is independentof the initial iterative point, so Comprehensive Algorithm B is naturally independent of the initialiterative point.

Theorem 3. Comprehensive Integrated Algorithm A is independent of the initial iterative point.

413


Proof. Comprehensive Integrated Algorithm A consists of two parts: the hybrid second orderalgorithm and Comprehensive Algorithm A. Whether the test point is very far or very close to theplanar algebraic curve, the hybrid second order algorithm is executed several times. If this algorithmconverges, then it represents that the execution of Comprehensive integrated Algorithm A is over.So Comprehensive Integrated Algorithm A is independent of the initial iterative point. If the hybridsecond order algorithm does not converge, then run Comprehensive Algorithm A of the second stepof Comprehensive Integrated Algorithm A. Because whether the test point is very far from or veryclose to the planar algebraic curve, we know from Theorem 1 that Comprehensive Algorithm A isindependent of the initial iterative point. To sum up, Comprehensive Integrated Algorithm A isindependent of the initial iterative point.

In a similar way to the proof of Theorem 3, we can state the following result.

Theorem 4. Comprehensive Integrated Algorithm B is independent of the initial iterative point.

Theorem 5. The first remedial algorithm of Comprehensive Algorithm B is independent of the initialiterative point.

Proof. From the Figure 5, for any initial iterative point, the final iterative point pc of Algorithm 1 in thefirst step of the first remedial algorithm of Comprehensive Algorithm B can ensure that it falls on theplanar algebraic curve f (x). The left endpoint L0 is the only one that can be determined through thirdstep of the first remedial algorithm of Comprehensive Algorithm B. Graphic display shows that theleft endpoint L0 and the original test point p are on both sides of the planar algebraic curve. Namely,there is only one intersection point (also written as pc) between the line segment L0p and the planaralgebraic curve f (x). Because the hybrid method of combining Newton’s iterative method and binarysearch method is global convergence method, the intersection pc of the line segment L0p and theplanar algebraic curve f (x) can be accurately and uniquely solved by this method. Then repeatedlyiterate and run Step 2, Step 3 and Step 4, the distance ‖pc − pΓ‖ between the current intersection pointpc and the orthogonal projection point pΓ of the original test point p continues to shrink to zero. So wehave this conclusion that the first remedial algorithm of Comprehensive Algorithm B is independentof the initial iterative point.

Theorem 6. Second Remedial Algorithm is independent of the initial iterative point.

Proof. In Remark 3, we give a detailed description of the geometric interpretation of Second RemedialAlgorithm. In this proof, we only elaborate on the most important geometric significance of SecondRemedial Algorithm. The first step of Second Remedial Algorithm is to let the initial iteration point fallon the planar algebraic curve as much as possible through Newton’s steepest gradient descent methodof Algorithm 1. Moreover, there is few restriction on the selection of the initial iterative point. Thepurpose of Do . . . While cycle body in Second Remedial Algorithm is to continuously and graduallymove the iterative point pc to fall on the planar algebraic curve to the orthogonal projection pointpΓ. The second step in Do. . . While cycle body in Second Remedial Algorithm has two characteristics.Since the footpoint q is the unique intersection point of the curvature circle and the straight linesegment mp connecting the centre m of the curvature circle and the test point p, the footpoint q isalways closely related to the iterative point pc fallen on the planar algebraic curve and the test pointp. The first characteristic is that the footpoint q can guarantee the global convergence of the wholealgorithm (Second Remedial Algorithm). The second characteristic is that the distance between thefootpoint q and the planar algebraic curve is much smaller than the distance between the test pointp and the planar algebraic curve. So the third step with Algorithm 1 in Do . . . While cycle bodycan very robustly iterate the footpoint q onto the planar algebraic curve. The core thought of Do

. . . While cycle body in Second Remedial Algorithm is to keep the iterative point pc fallen on the planar

414


algebraic curve moving towards the orthogonal projection point pΓ such that the distance ‖pc − pΓ‖between the iterative point pc and the orthogonal projection point pΓ becomes smaller and smaller.If the distance ‖pc − pΓ‖ gets smaller and smaller, we have found that the decreasing speed of thedistance ‖pc − pΓ‖ is getting slower and slower. Especially the second formula of the formula (8) isvery difficult to be satisfied. Algorithm 12 behind the loop body has four advantages and importantroles: (1) Algorithm 12 plays a role for accelerating the whole algorithm (Second Remedial Algorithm);(2) The first step plays a role for making the iteration point fall on the planar algebraic curve as faras possible; (3) The second step plays a role for accelerating orthogonalization between the vector−→ppc and the vector ∇ f (pc); (4) The third step plays a dual role for the accelerating orthogonalizationand the promoting the iterative point to fall on the planar algebraic curve. No matter how far the testpoint is from the planar algebraic curve, Second Remedial Algorithm converges very robustly andefficiently. By adding this step, the efficiency and the robustness for Algorithm 12 of Second RemedialAlgorithm is further improved. Then the robustness and the efficiency of Second Remedial Algorithmis also further improved. So Second Remedial Algorithm is independent of the initial iterative point.In addition, in a similar way to the proof of Theorem 3, it is not difficult to know that ComprehensiveIntegrated Algorithm C is also independent of the initial iterative point.

4. Numerical Comparison Results

We now present some examples to illustrate the efficiency and the comparison of the newlydeveloped method of Comprehensive Algorithm B and Second Remedial Algorithm to show the twoalgorithms’ high robustness and efficiency for very remote test points. We have three examples torepresent closed planar algebraic curve, two sub-closed planar algebraic curves, two branches butnot closed planar algebra curves and a single branch not closed the planar algebra curve, respectively.All computations were done using VC++6.0. We used ε = 10−16. The following stopping criteriais used for Comprehensive Algorithm B and Second Remedial Algorithm . In Tables 1–3, the foursymbols p, pΓ, | f (pΓ)| and |〈V1, V2〉| are the original test point, the orthogonal projection point of theoriginal test point, the deviation degree of the orthogonal projection point on the planar algebraiccurve and the absolute value of the inner product of two vectors V1 and V2, respectively, where V1

is −→ppΓ and V2 is the tangent vector[− ∂ f

∂y , ∂ f∂x

]of the orthogonal projection point pΓ on the planar

algebraic curve f (x). Thanks to the suggestions by the reviewers, the fourth quadrant result values ofthe three tables are implemented by Second Remedial Algorithm in Maple 18 environment.

Example 1 (Reference to [28]). Suppose a planar algebraic curvef (x, y) = x6 + 2x5y− 2x3y2 + x4 − y3 + 2y8 − 4 = 0 (See Figure 7). In each of the four quadrants,randomly select four distant test points. We calculate the corresponding orthogonal projection point for each testpoint via computation by Comprehensive Algorithm B and Second Remedial Algorithm. The specific results areshown in Table 1).

Example 2. Suppose a planar algebraic curve f (x, y) = x10 + 6xy + 2y18 − 2 = 0(See Figure 8). In each ofthe four quadrants, randomly select four distant test points. We calculate the corresponding orthogonal projectionpoint for each test point via computation by Comprehensive Algorithm B and Second Remedial Algorithm. Thespecific results are shown in Table 2.

Example 3. Suppose a planar algebraic curve f (x, y) = x10 + 6xy + 2y16 + 2 = 0(See Figure 9). In each ofthe four quadrants, randomly select four distant test points. We calculate the corresponding orthogonal projectionpoint for each test point via computation by Comprehensive Algorithm B and Second Remedial Algorithm. Thespecific results are shown in Table 3.

415


Figure 7. Graphical representation of the planar algebraic curve for Example 1.



416


Ta

ble

1.

Test

resu

lts

ofC

ompr

ehen

sive

Alg

orit

hmB

for

Exam

ple

1.

p(1

325,

7447

)(7

79,3

25)

(990

,137

5)(0

.59,

1377

)(−

0.5,

8623

)(−

16,5

98)

(−3,

231)

(−21

,247

)

pΓ

(3.1

087,−1

.866

6)(3

.364

7,−1

.818

4)(3

.369

5,−1

.838

0)(0

.60,

1.14

0)(−

0.01

582,

1.13

368)

(−0.

3353

,1.1

3074

)(−

0.22

68,1

.132

8)(−

0.60

71,1

.116

6)|f(

pΓ) |

3.9·1

0−10

7.9·1

0−11

1.4·1

0−11

2.7·1

0−12

2.3·1

0−12

4.0·1

0−12

1.9·1

0−12

2.3·1

0−12

|〈V1,

V2〉|

2.3·1

0−10

4.4·1

0−11

2.9·1

0−09

4.8·1

0−12

2.1·1

0−13

1.5·1

0−12

1.4·1

0−11

9.6·1

0−12

p(−

42,−

127)

(−5,−3

8)(−

9,−5

79)

(−53

7,−1

1)(7

8,−1

23)

(168

,−12

)(5

37,−

31)

(91,−2

21)

pΓ

(0.6

633,

-0.9

941)

(0.4

951,−1

.029

2)(−

0.21

83,−

1.04

45)

(−1.

2752

,0.7

396)

(3.3

519,−1

.868

5)(3

.369

4,−1

.841

7)(3

.369

4,−1

.841

4)(3

.341

5,−1

.873

6)|f(

pΓ) |

7.4·1

0−12

1.4·1

0−12

1.7·1

0−12

6.1·1

0−12

4.9·1

0−12

1.8·1

0−13

5.4·1

0−13

8.6·1

0−12

|〈V1,

V2〉|

1.7·1

0−13

1.2·1

0−12

1.1·1

0−12

1.7·1

0−12

2.3·1

0−13

5.7·1

0−14

2.8·1

0−13

4.5·1

0−12

Ta

ble

2.

Test

resu

lts

ofC

ompr

ehen

sive

Alg

orit

hmB

for

Exam

ple

2.

p(5

65,9

45)

(979

,325

)(3

75,4

05)

(195

9,13

77)

(−93

56,8

623)

(−81

6,79

8)(−

3987

,123

1)(−

4821

,647

)

pΓ

(0.2

978,−0

.910

8)

(1.0

773,−0

.016

4)

(0.6

094,

0.54

50)

(0.4

780,

0.69

61)

(−1.

2055

,1.0

209

)(−

1.20

35,1

.023

0)

(−1.

2288

,0.9

794

)(−

1.23

41,0

.955

4)

|f(p

Γ) |

1.1·1

0−14

6.0·1

0−13

8.0·1

0−18

9.2·1

0−15

5.8·1

0−15

1.1·1

0−14

2.5·1

0−15

1.1·1

0−14

|〈V1,

V2〉|

01.

8·1

0−12

2.3·1

0−13

4.5·1

0−12

6.9·1

0−10

2.1·1

0−11

1.7·1

0−10

1.1·1

0−10

p(−

7942

,−27

5)(−

598,−9

8)(−

3709

,−19

79)

(−29

37,−

1391

)(9

708,−3

23)

(260

8,−1

912)

(734

7,−9

31)

(509

1,−1

921)

pΓ

(−1.

2367

,0.8

930

)(−

1.18

47,0

.485

1)

(−1.

0035

,−0.

1601

)(−

1.02

25,0

.122

4)

(1.1

427,−0

.910

0)(1

.122

4,−0

.978

7)(1

.141

4,−0

.927

3)(1

.134

4,−0

.956

3)|f(

pΓ) |

2.3·1

0−15

1.3·1

0−11

9.5·1

0−15

1.4·1

0−13

4.7·1

0−15

3.2·1

0−15

4.4·1

0−15

1.7·1

0−17

|〈V1,

V2〉|

4.0·1

0−11

3.6·1

0−12

8.7·1

0−11

7.2·1

0−12

5.8·1

0−11

8.7·1

0−11

7.2·1

0−12

2.1·1

0−10

Ta

ble

3.

Test

resu

lts

ofC

ompr

ehen

sive

Alg

orit

hmB

for

Exam

ple

3.

p(1

387,

645)

(187

9,39

5)(3

075,

205)

(195

6,77

7)(−

9256

,460

3)(−

836,

1798

)(−

5987

,103

1)(−

4181

,124

7)

pΓ

(−1.

0629

,0.6

023)

(0.4

232,−0

.865

0)(0

.421

4,−0

.851

5)(0

.427

6,−0

.879

5)(−

1.13

05,0

.965

5)

(−1.

0823

,1.0

103

)(−

0.42

32,0

.822

0)(−

1.13

69,0

.949

0)

|f(p

Γ) |

1.7·1

0−13

1.3·1

0−16

8.1·1

0−17

1.0·1

0−16

5.3·1

0−17

1.8·1

0−15

6.1·1

0−17

1.0·1

0−17

|〈V1,

V2〉|

3.6·1

0−12

3.2·1

0−12

6.3·1

0−13

6.3·1

0−12

1.7·1

0−10

4.3·1

0−11

1.8·1

0−12

6.5·1

0−11

p(−

7342

,−12

75)

(−50

98,−

918)

(−32

17,−

2079

)(−

2337

,−12

51)

(950

8,−3

75)

(660

8,−7

12)

(234

7,−9

31)

(149

1,−1

321)

pΓ

(−0.

4226

,0.8

617

)(−

0.42

27,0

.862

3)

(−1.

0274

,0.5

370)

(−1.

0471

,0.5

706)

(1.2

367,−0

.926

7)(1

.235

3,−0

.946

0)(1

.225

5,−0

.988

8)(1

.206

8,−1

.019

4)|f(

pΓ) |

5.7·1

0−17

1.3·1

0−16

4.4·1

0−14

1.1·1

0−13

2.2·1

0−15

2.8·1

0−15

1.8·1

0−15

1.1·1

0−15

|〈V1,

V2〉|

1.0·1

0−11

5.5·1

0−12

7.3·1

0−12

9.1·1

0−12

2.2·1

0−11

3.6·1

0−12

5.4·1

0−11

2.2·1

0−11

417


Remark 5. Besides all test points of the three examples mentioned above are tested by Comprehensive AlgorithmB, we have tested them again with the Second Remedial Algorithm. All the test results are consistent with thoseof Comprehensive Algorithm B and convergent. In addition, in the region [−3000, 3000]× [−3000, 3000] ofeach example, we randomly select a large number of test points, the probability of non-convergence is particularlylow by Second Remedial Algorithm. Further, we use Second Remedial Algorithm other examples with test pointsin a very large area, and the probability of non-convergence is also very low. Second Remedial Algorithm isverified to be the best one again in our paper. Of course, the robustness and the efficiency of Second RemedialAlgorithm is better than that of the existing algorithms.

5. Conclusions and Future Work

In this paper, we have constructed a Comprehensive Algorithm which is an improved curvaturecircle algorithm for orthogonal projecting onto planar algebraic curve. Based on an integrated hybridsecond-order algorithm [28], the Comprehensive Algorithm (the improved curvature circle algorithm)has also incorporated the curvature circle technique and Newton’s gradient steepest descent methodsuch that it can converge robustly and efficiently no matter how far the test point is from the planaralgebraic curve and no matter where the initial iterative point is located. Furthermore, we proposeSecond Remedial Algorithm based on Comprehensive Algorithm B. In particular, its robustnessand efficiency is greatly improved than that of Comprehensive Algorithm B and it achieves ourexpected result. The numerical examples show that the improved curvature circle algorithm is superiorto the existing ones. In future work, we try to refine the idea of Comprehensive Algorithm and SecondRemedy Algorithm. And the idea is applied to point orthogonal projecting onto spatial algebraic curveand algebraic surface.

Author Contributions: The contribution of all the authors is the same. All of the authors team up to developthe current draft. Z.W. is responsible for investigating, providing resources and methodology, the original draft,writing, reviewing, validation, editing and supervision of this work. X.L. is responsible for software, algorithm,program implementation, formal analysis, visualization, writing, reviewing, editing and supervision of this work.

Funding: This research was funded by the National Natural Science Foundation of China Grant No. 61263034,the Feature Key Laboratory for Regular Institutions of Higher Education of Guizhou Province Grant No. 2016003,the Key Laboratory of Advanced Manufacturing Technology, Ministry of Education, Guizhou University GrantNo. 2018479, the National Bureau of Statistics Foundation Grant No. 2014LY011, the Shandong Provincial NaturalScience Foundation of China Grant No.ZR2016GM24.

Acknowledgments: We take the opportunity to thank the anonymous reviewers for their thoughtful andmeaningful comments.


References

1. Bajaj, C.L.; Bernardini, F.; Xu, G. Reconstructing surfaces and functions on surfaces from unorganizedthree-dimensional data. Algorithmica 1997, 19, 243–261. [CrossRef]

2. Juttler, B.; Felis, A. Least-squares fitting of algebraic spline surfaces. Adv. Comput. Math. 2002, 17, 135–152.3. Song, X.; Juttler, B. Modeling and 3D object reconstruction by implicitly defined surfaces with sharp features.

Comput. Graph. 2009, 33, 321–330. [CrossRef]4. Schulz, T.; Juttler, B. Envelope computation in the plane by approximate implicitization. Appl. Algebra Eng.

Commun. Comput. 2011, 22, 265–288. [CrossRef]5. Kanatani, K.; Sugaya, Y. Unified computation of strict maximum likelihood for geometric fitting. J. Math.

Imaging Vis. 2010, 38, 1–13. [CrossRef]6. Mullen, P.; De Goes, F.; Desbrun, M.; Cohen-Steiner, D.; Alliez, P. Signing the unsigned: Robust surface

reconstruction from raw pointsets. Comput. Graph. Forum 2010, 29, 1733–1741. [CrossRef]7. Upreti, K.; Subbarayan, G. Signed algebraic level sets on NURBS surfaces and implicit Boolean signed

algebraic level sets on NURBS surfaces and implicit Boolean. Comput. Aided Des. 2017, 82,112–126. [CrossRef]8. Rouhani, M., Sappa, A.D. Implicit polynomial representation through a fast fitting error estimation.

IEEE Trans. Image Process. 2012, 21, 2089–2098. [CrossRef]

418


9. Zagorchev, L.G.; Goshtasby, A.A. A curvature-adaptive implicit surface reconstruction for irregularlyspaced points. IEEE Trans. Vis. Comput. Graph. 2012, 18, 1460–1473. [CrossRef]

10. William, H.P.; Brian, P.F.; Teukolsky, S.A.; William, T.V. Numerical Recipes in C: The Art of Scientific Computing,2nd ed.; Cambridge University Press: Cambridge, UK, 1992.

11. Steve, S.; Sandford, L.; Ponce, J. Using geometric distance fits for 3-D object modeling and recognition.IEEE Trans. Pattern Anal. Mach. Intell. 1994, 16, 1183–1196.

12. Morgan, A.P. Polynomial continuation and its relationship to the symbolic reduction of polynomial systems.In Symbolic and Numerical Computation for Artificial Intelligence; Academic Press: Cambridge, MA, USA, 1992;pp. 23–45.

13. Layne, T.W.; Billups, S.C.; Morgan, A.P. Algorithm 652: HOMPACK: A suite of codes for globally convergenthomotopy algorithms. ACM Trans. Math. Softw. 1987, 13, 281–310.

14. Berthold, K.P.H. Relative orientation revisited. J. Opt. Soc. Am. A 1991, 8, 1630–1638.15. Dinesh, M.; Krishnan, S. Solving algebraic systems using matrix computations. ACM Sigsam Bull. 1996,

30, 4–21.16. Chionh, E.-W. Base Points, Resultants, and the Implicit Representation of Rational Surfaces. Ph.D. Thesis,

University of Waterloo, Waterloo, ON, Canada, 1990.17. De Montaudouin, Y.; Tiller, W. The Cayley method in computer aided geometric design. Comput. Aided

Geom. Des. 1984, 1, 309–326. [CrossRef]18. Albert, A.A. Modern Higher Algebra; D.C. Heath and Company: New York, NY, USA, 1933.19. Thomas, W.; David, S.; Anderson, C.; Goldman, R.N. Implicit representation of parametric curves

and surfaces. Comput. Vis. Graph. Image Proc. 1984, 28, 72–84.20. Nishita, T.; Sederberg, T.W.; Kakimoto, M. Ray tracing trimmed rational surface patches. ACM SIGGRAPH

Comput. Graph. 1990, 24, 337–345. [CrossRef]21. Elber, G.; Kim, M.-S. Geometric Constraint Solver Using Multivariate Rational Spline Functions.

In Proceedings of the 6th ACM Symposium on Solid Modeling and Applications, Ann Arbor, MI, USA,4–8 June 2001; pp. 1–10.

22. Sherbrooke, E.C.; Patrikalakis, N.M. Computation of the solutions of nonlinear polynomial systems.Comput. Aided Geom. Des. 1993, 10, 379–405. [CrossRef]

23. Hartmann, E. The normal form of a planar curve and its application to curve design. In Mathematical Methodsfor Curves and Surfaces II; Vanderbilt University Press: Nashville, TN, USA, 1997; pp. 237–244.

24. Hartmann, E. On the curvature of curves and surfaces defined by normal forms. Comput. Aided Geom. Des.1999, 16, 355–376. [CrossRef]

25. Nicholas, J.R. Implicit polynomials, orthogonal distance regression, and the closest point on a curve.IEEE Trans. Pattern Anal. Mach. Intell. 2000, 22, 191–199.

26. Martin, A.; Bert, J. Robust computation of foot points on implicitly defined curves. In Mathematical Methodsfor Curves and Surfaces: Troms? Nashboro Press: Brentwood, TN, USA, 2004; pp. 1–10.

27. Hu, M.; Zhou, Y.; Li, X. Robust and accurate computation of geometric distance for Lipschitz continuousimplicit curves. Vis. Comput. 2017, 33, 937–947. [CrossRef]

28. Li, X.; Pan, F.; Cheng, T.; Wu, Z.; Liang, J.; Hou, L. Integrated hybrid second order algorithm for orthogonalprojection onto a planar implicit curve. Symmetry 2018, 10, 164. [CrossRef]

29. Goldman R. Curvature formulas for implicit curves and surfaces. Comput. Aided Geom. Des. 2005, 22, 632–658.[CrossRef]


419

mathematics

Article

Nonlinear Operators as Concerns ConvexProgramming and Applied to Signal Processing

Anantachai Padcharoen 1 and Pakeeta Sukprasert 2,*1 Department of Mathematics, Faculty of Science and Technology, Rambhai Barni Rajabhat University,

Chanthaburi 22000, Thailand; [email protected] Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala

University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand* Correspondence: [email protected]

Received: 16 August 2019; Accepted: 13 September 2019; Published: 19 September 2019

Abstract: Splitting methods have received a lot of attention lately because many nonlinear problemsthat arise in the areas used, such as signal processing and image restoration, are modeled inmathematics as a nonlinear equation, and this operator is decomposed as the sum of two nonlinearoperators. Most investigations about the methods of separation are carried out in the Hilbert spaces.This work develops an iterative scheme in Banach spaces. We prove the convergence theorem of ouriterative scheme, applications in common zeros of accretive operators, convexly constrained leastsquare problem, convex minimization problem and signal processing.

Keywords: convexity; least square problem; accretive operators; signal processing

1. Introduction

Let E be a real Banach space. The zero point problem is as follows:

find x ∈ E such that 0 ∈ Au +Bu, (1)

where A : E → E is an operator and B : E → 2E is a set-valued operator. This problemincludes, as special cases, convex programming, variational inequalities, split feasibility problemand minimization problem [1–7]. To be more precise, some concrete problems in machine learning,image processing [4,5], signal processing and linear inverse problem can be modeled mathematicallyas the form in Equation (1).

Signal processing and numerical optimization are independent scientific fields that have alwaysbeen mutually influencing each other. Perhaps the most convincing example where the two fieldshave met is compressed sensing (CS) [2]. Several surveys dedicated to these algorithms and theirapplications in signal processing have appeared [3,6–8]

Fixed point iterations is an important tool for solving various problems and is known in a Banachspace E. Let K be a nonempty closed convex subset of E and S : K→ K is the operator with at leastone fixed point. Then, for u1 ∈ K :

1. The Picard iterative scheme [9] is defined by:

un+1 = Sun, ∀ n ∈ N.

2. The Mann iterative scheme [10] is defined by:

un+1 = (1− ηn)un + ηnSun, ∀ n ∈ N,

where {ηn} is a sequence in (0, 1).



3. The Ishikawa iterative scheme [11] is defined by:

un+1 = (1− ηn)un + ηnS[(1− ϑn)un + ϑnSun], ∀ n ∈ N,

where {ηn} and {ϑn} are sequences in (0, 1).4. The S-iterative scheme [12] is defined by:

un+1 = (1− ηn)Sun + ηnS[(1− ϑn)un + ϑnSun], ∀ n ∈ N,

where {ηn} and {ϑn} are sequences in (0, 1).

Recently, Sahu et al. [13] and Thakur et al. [14] introduced the following same iterative scheme fornonexpansive mappings in uniformly convex Banach space:⎧⎪⎪⎨⎪⎪⎩

wn = (1− ξn)un + ξnSun,

zn = (1− ϑn)wn + ϑnSwn,

un+1 = (1− ηn)Swn + ηnSzn, ∀ n ∈ N,

where {ηn}, {ϑn} and {ξn} are sequences in (0, 1). The authors proved that this scheme converges toa fixed point of contraction mapping, faster than all known iterative schemes. In addition, the authorsprovided an example to support their claim.

In this paper, we first develop an iterative scheme for calculating common solutions and usingour results to solve the problem in Equation (1). Secondly, we find common solutions of convexlyconstrained least square problems, convex minimization problems and applied to signal processing.

2. Preliminaries

Let E be a real Banach space with norm ‖ · ‖ and E∗ be its dual. The value of f ∈ E∗ at u ∈ E

ia denoted by 〈u, f 〉. A Banach space E is called strictly convex if ‖u+v‖2 < 1, for all u, v ∈ E with

‖u‖ = ‖v‖ = 1. It is called uniformly convex if limn→∞ ‖un − vn‖ = 0 for any two sequences{un}, {vn} in E such that ‖u‖ = ‖v‖ = 1 and limn→∞

‖u+v‖2 = 1.

The (normalized) duality mapping J from E into the family of nonempty (by Hahn Banachtheorem) weak-star compact subsets of its dual E is defined by

J (u) = { f ∈ E∗ : 〈u, f 〉 = ‖u‖2 = ‖ f ‖2}

for each u ∈ E, where 〈·, ·〉 denotes the generalized duality pairing.For an operator A : E→ 2E, we denote its domain, range and graph as follows:

D(A) = {u ∈ E : Au �= ∅}R(A) = ∪{Ap : p ∈ D(A)},

andG(A) = {(u, v) ∈ E× E : u ∈ D(A), v ∈ Au},

respectively. The inverse A−1 of A is defined by u ∈ A−1v, if and only if v ∈ Au. If ∀ui ∈ D(A) andvi ∈ Aui (i = 1, 2), and there is j ∈ J (u1 − u2) such that 〈v1 − v2, j〉 ≥ 0, then A is called accretive.

An accretive operator A in a Banach space E is said to satisfy the range condition ifD(A) ⊂ R(I + μA) for all μ > 0, where D(A) denotes the closure of the domain of A. We knowthat for an accretive operator A which satisfies the range condition, A−10 = Fix(JA

μ ) for all μ > 0.A point u ∈ K is a fixed point of S provided Su = u. Denote by Fix(S) the set of fixed points of S,

i.e., Fix(S) = {u ∈ K : Su = u}.

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1. The mapping S is called L−Lipschitz, L > 0, if

‖Su− Sv‖ ≤ L‖u− v‖, ∀ u, v ∈ K.

2. The mapping S is called nonexpansive if

‖Su− Sv‖ ≤ ‖u− v‖, ∀ u, v ∈ K.

3. The mapping S is called quasi-nonexpansive if Fix(S) �= ∅ and

‖Su− v‖ ≤ ‖u− v‖, ∀ u ∈ K, v ∈ Fix(S).

In this case, H is a real Hilbert space. If A : E→ 2E is an m−accretive operator (see [15–17]), thenA is called maximal accretive operator [18], and for all μ > 0,R(I + μA) = H if and only if A is calledmaximal monotone [19]. Denote by dom(h) the domain of a function h : H→ (−∞, ∞], i.e.,

dom(h) = {u ∈ H : h(u) < ∞}.

The subdifferential of h ∈ Γ0(H) at u ∈ H is the set

∂h(u) = {z ∈ H : h(u) ≤ h(v) + 〈z, u− v〉, ∀ v ∈ H},

where Γ0(H) denotes the class of all l.s.c. functions from H to (−∞, ∞] with nonempty domains.

Lemma 1 ([20]). Let h ∈ Γ0(H). Then, ∂h is maximal monotone.

We denote by Bλ[v] the closed ball with the center at v and radius λ :

Bλ[v] = {u ∈ E : ‖v− u‖ ≤ λ}.

Lemma 2 ([21]). Let E be a Banach space, and p > 1 and R > 0 be two fixed numbers. Then, E is uniformlyconvex if and only if there exists a continuous, strictly increasing, and convex function ϕ : [0, ∞) → [0, ∞)

with ϕ(0) = 0 such that

‖αu + (1− α)v‖p ≤ ‖u‖p + (1− α)‖v‖p − α(1− α)ϕ(‖u− v‖),

for all u, v ∈ BR[0] and α ∈ [0, 1].

Definition 1 ([22]). A vector space H is said to satisfy Opial’s condition, if for each sequence {un} in H whichconverges weakly to point u ∈ H,

lim infn→∞

‖un − u‖ < lim infn→∞

‖un − v‖, v ∈ H, v �= u.

Lemma 3 ([23]). Let K be a nonempty subset of a Banach space E, let S : K→ E be a uniformly continuousmapping, and let {un} ⊂ K an approximating fixed point sequence of S. Then, {vn} is an approximating fixedpoint sequence of S whenever {vn} is in K such that limn→∞ ‖un − vn‖ = 0.

Lemma 4 ([16]). Let K be a nonempty closed convex subset of a uniformly convex Banach space E. If S : K→ E

is a nonexpansive mapping, then I − S has the demiclosed property with respect to 0.

A subset K of Banach space E is called a retract of E if there is a continuous mapping Q from E

onto K such that Qu = u for all u ∈ K. We call such Q a retraction of E onto K. It follows that, if amapping Q is a retraction, then Qv = v for all v in the range of Q. A retraction Q is called a sunny if

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Q(Qu + λ(u− Qu)) = Qu for all u ∈ E and λ ≥ 0. If a sunny retraction Q is also nonexpansive, then K

is called a sunny nonexpansive retract of E [24].Let E be a strictly convex reflexive Banach space and K be a nonempty closed convex subset of E.

Denote by PK the (metric) projection from E onto K, namely, for u ∈ E, PK(u) is the unique point inK with the property

inf{‖u− v‖ : v ∈ K} = ‖u− PK(u)‖.

Let an inner product 〈·, ·〉 and the induced norm ‖ · ‖ are specified with a real Hilbert space H.Let K is a nonempty subset of H, we have the nearest point projection PK : H → K is the uniquesunny nonexpansive retraction of H onto K. It is also known that PK(u) ∈ K and

〈u− PK(u),PK(u)− v〉 ≥ 0, ∀ u ∈ H, v ∈ K.

3. Main Results

Let K be a nonempty closed convex subset of a Banach space E with QK as a sunny nonexpansiveretraction. We denote by Ψ := Fix(S) ∩ Fix(T).

Lemma 5. Let K be a nonempty closed convex subset of a Banach space E with QK as the sunny nonexpansiveretraction, let S,T : K→ E be quasi-nonexpansive mappings which Ψ �= ∅, and let {ηn}, {ϑn} and {ξn} besequences in (0, 1) for all n ∈ N. Let {un} be defined by Algorithm 1. Then, for each u ∈ Ψ, limn→∞ ‖un − u‖exists and

‖wn − u|| ≤ ‖un − u‖, and ‖zn − u‖ ≤ ‖un − u‖, ∀ n ∈ N. (2)

Algorithm 1: Three-step sunny nonexpansive retraction

initialization: ηn, ϑn, ξn ∈ (0, 1), u1 ∈ K and n = 1.while stopping criterion not met do

wn = QK[(1− ξn)un + ξnSun],zn = QK[(1− ϑn)wn + ϑnTwn],un+1 = QK[(1− ηn)Swn + ηnTzn].

end

Proof. Let u ∈ Ψ. Then, we have

‖wn − u‖ = ‖QK[(1− ξn)un + ξnSun]− u‖≤ ‖(1− ξn)(un − u) + ξn(Sun − u)‖≤ (1− ξn)‖un − u‖+ ξn‖Sun − u‖≤ (1− ξn)‖un − u‖+ ξn‖un − u‖= ‖un − u‖,

(3)

‖zn − u‖ = ‖QK[(1− ϑn)wn + ϑnTwn]− u‖≤ ‖(1− ϑn)(wn − u) + ϑn(Twn − u)‖≤ (1− ϑn)‖wn − u‖+ ϑn‖Twn − u‖≤ (1− ϑn)‖wn − u‖+ ϑn‖wn − u‖= ‖wn − u‖≤ ‖un − u‖,

(4)

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and‖un+1 − u‖ = ‖QK[(1− ηn)Swn + ηnTzn]− u‖

≤ ‖(1− ηn)(Swn − u) + ηn(Tzn − u)‖≤ (1− ηn)‖Swn − u‖+ ηn‖Tzn − u‖≤ (1− ηn)‖wn − u‖+ ηn‖zn − u‖≤ (1− ηn)‖un − u‖+ ηn‖un − u‖= ‖un − u‖.

(5)

Therefore,‖un+1 − u‖ ≤ ‖un − u‖ ≤ · · · ≤ ‖u1 − u‖, ∀ n ∈ N. (6)

Since {‖un − u‖} is monotonically decreasing, we have that the sequence {‖un − u‖}is convergent.

From Lemma 5, we have results:

Theorem 1. Let K be a nonempty closed convex subset of a Banach space E with QK as the sunny nonexpansiveretraction, let S,T : K → E be quasi-nonexpansive mappings with Ψ �= ∅, and let {ηn}, {ϑn} and {ξn} besequences of real numbers, for which 0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤ ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1for all n ∈ N. Let u1 ∈ K, PΨ(u1) = u∗ and {un} is defined by Algorithm 1. Then, we have the following:

(i) {un} is in a closed convex bounded set Bλ[u∗] ∩ K, where λ is a constant in (0, ∞) such that‖u1 − u∗‖ ≤ λ.

(ii) If S is uniformly continuous, then limn→∞ ‖un − Sun‖ = 0 and limn→∞ ‖un − Tun‖ = 0.(iii) If E fulfills the Opial’s condition and I − S and I − T are demiclosed at 0, then {un} converges weakly to an

element of Ψ ∩ Bλ[u∗].

Proof. (i) Since u∗ ∈ Ψ, from Equation (6), we obtain

‖un+1 − u∗‖ ≤ ‖un − u∗‖ ≤ · · · ≤ ‖u1 − u∗‖ ≤ λ, ∀ n ∈ N. (7)

Therefore, {un} is in the closed convex bounded set Bλ[u∗] ∩K.

(ii) Suppose that S is uniformly continuous. Using Lemma 5, we get that {un}, {zn} and {wn}are in Bλ[u∗] ∩K, and hence, from Equation (2), we obtain

‖Twn − u∗‖ ≤ λ, ‖Swn − u∗‖ ≤ λ and ‖Sun − u∗‖ ≤ λ, ∀ n ∈ N.

Using Lemma 2 for p = 2 and R = λ, from Equation (5), we obtain

‖un+1 − u∗‖2 ≤ ‖(1− ηn)(Swn − u∗) + ηn(Tzn − u∗)‖2

≤ (1− ηn)‖Swn − u∗‖2 + ηn‖Tzn − u∗‖2

− ηn(1− ηn)ϕ(‖Swn − Tzn‖)≤ (1− ηn)‖wn − u∗‖2 + ηn‖zn − u∗‖2

− ηn(1− ηn)ϕ(‖Swn − Tzn‖)≤ (1− ηn)‖un − u∗‖2 + ηn‖un − u∗‖2

− ηn(1− ηn)ϕ(‖Swn − Tzn‖)= ‖un − u∗‖2 − ηn(1− ηn)ϕ(‖Swn − Tzn‖),

(8)

which implies that

ηn(1− ηn)ϕ(‖Swn − Tzn‖) = ‖un − u∗‖ − ‖un+1 − u∗‖2. (9)

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Note that: c1(1− c1) ≤ ηn(1− ηn). Thus,

c1(1− c1)n

∑i=1

ϕ(‖Swi − Tzi‖) = ‖u1 − u∗‖ − ‖un+1 − u∗‖2, ∀ n ∈ N. (10)

In the same way, we obtain

c1(1− c1)∞

∑n=1

ϕ(‖Swn − Tzn‖) ≤ ‖u1 − u∗‖ < ∞. (11)

Therefore, we have limn→∞ ‖Swn − Tzn‖ = 0. From the relations in Algorithm 1, we obtain

‖wn − u∗‖2 ≤ (1− ξn)‖un − u∗‖2 + ξn‖Sun − u∗‖2

− ξn(1− ξn)ϕ(‖un − Sun‖)≤ (1− ξn)‖un − u∗‖2 + ξn‖un − u∗‖2

− ξn(1− ξn)ϕ(‖un − Sun‖)= ‖un − u∗‖2 − ξn(1− ξn)ϕ(‖un − Sun‖)

(12)

and‖zn − u∗‖2 ≤ ‖(1− ϑn)(wn − u∗) + ϑn(Twn − u∗)‖2

≤ (1− ϑn)‖wn − u∗‖2 + ϑn‖Twn − u∗‖2

− ϑn(1− ϑn)ϕ(‖wn − Twn‖)≤ (1− ϑn)‖wn − u∗‖2 + ϑn‖wn − u∗‖2

= ‖wn − u∗‖2 − ϑn(1− ϑn)ϕ(‖wn − Twn‖)≤ ‖un − u∗‖2 − ϑn(1− ϑn)ϕ(‖wn − Twn‖).

(13)

From Equations (8), (13) and (12), we obtain

‖un+1 − u∗‖2 ≤ ‖(1− ηn)(Swn − u∗) + ηn(Tzn − u∗)‖2

≤ (1− ηn)‖Swn − u∗‖2 + ηn‖Tzn − u∗‖2

− ηn(1− ηn)ϕ(‖Swn − Tzn‖)≤ (1− ηn)‖wn − u∗‖2 + ηn‖zn − u∗‖2

− ηn(1− ηn)ϕ(‖Swn − Tzn‖)≤ (1− ηn)[‖un − u∗‖2 − ξn(1− ξn)ϕ(‖un − Sun‖)]+ ηn[‖un − u∗‖2 − ϑn(1− ϑn)ϕ(‖wn − Twn‖)]− ηn(1− ηn)ϕ(‖Swn − Tzn‖)

= ‖un − u∗‖2 − (1− ηn)ξn(1− ξn)ϕ(‖un − Sun‖)− ηnϑn(1− ϑn)ϕ(‖wn − Twn‖)− ηn(1− ηn)ϕ(‖Swn − Tzn‖).

(14)

Note that: (1− c1)c3(1− c3) ≤ (1− ηn)ξn(1− ξn) and c1c2(1− c2) ≤ ηnϑn(1− ϑn). Thus,

(1− c1)c3(1− c3)n

∑i=1

ϕ(‖ui − Sui‖) ≤ ‖u1 − u∗‖2 − ‖un+1 − u∗‖2, ∀ n ∈ N.

It follows that limn→∞ ‖un − Sun|| = 0. Note that:

‖wn − un‖ = ‖QK[(1− ξn)un + ξnSun]− QK[un]‖≤ ‖Sun − un‖ → 0 as n → ∞.

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Since S is uniformly continuous, it follows from Lemma 3 that limn→∞ ‖wn − Swn‖ = 0.Thus, from limn→∞ ‖Swn − Tzn‖ = 0, we obtain limn→∞ ‖un − Tun‖ = 0.

(iii) By assumption, E satisfies the Opial’s condition. Let w∗ ∈ Ψ such that w∗ ∈ Bλ[u∗] ∩K.From Lemma 5, we have limn→∞ ‖un − w∗‖ exists. Suppose there are two subsequences {unq} and{uml} which converge to two distinct points u∗ and v∗ in Bλ[u∗] ∩K, respectively. Then, since bothI − S and I − T have the demiclosed property at 0, we have Su∗ = Tu∗ = u∗ and Sv∗ = Tv∗ = v∗.Moreover, using the Opial’s condition:

limn→∞

‖un − u∗‖ = limq→∞

‖unq − u∗‖ < liml→∞

‖uml − v∗‖ = limn→∞

‖un − v∗‖.

Similarly, we obtainlim

n→∞‖un − v∗‖ < lim

n→∞‖un − u∗‖,

which is a contradiction. Therefore, u∗ = v∗. Hence, the sequence {un} converges weakly to an elementof Ψ ∩ Bλ[u∗] ∩K.

Theorem 2. Let K be a nonempty closed convex subset of a Banach space E with QK as the sunny nonexpansiveretraction, let S,T : K→ E be nonexpansive mappings with Ψ �= ∅, and let {ηn}, {ϑn} and {ξn} be sequencesof real numbers, for which 0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤ ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1 for alln ∈ N. Let u1 ∈ K, PΨ(u1) = u∗ and {un} is defined by Algorithm 1. Then, we have the following:


(ii) limn→∞ ‖un − Sun‖ = 0 and limn→∞ ‖un − Tun‖ = 0.(iii) If E fulfills the Opial’s condition, then {un} converges weakly to an element of Ψ ∩ Bλ[u∗].

Proof. It follows from Theorem 1.

Corollary 1. Let K be a nonempty closed convex subset of a real Hilbert space H, let S,T : K → E benonexpansive mappings with Ψ �= ∅, and let {ηn}, {ϑn} and {ξn} be sequences of real numbers, for which0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤ ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1 for all n ∈ N. Let {un} be defined by⎧⎪⎪⎨⎪⎪⎩


zn = (1− ϑn)wn + ϑnTwn,

un+1 = (1− ηn)Swn + ηnTzn, ∀ n ∈ N.

(15)

Then, {un} converges weakly to an element of Ψ.

Proof. It follows from Theorem 1.

4. Applications

4.1. Common Zeros of Accretive Operators

From Equation (15), we set S = JAμ and T = JBμ , and inherit the convergence analysis for solving

Equation (1).

Theorem 3. Let K be a nonempty closed convex subset of a r.u.c. Banach space E satisfying theOpial’s condition. Let A : D(A) ⊆ K → 2E, B : D(B) ⊆ K → 2E be accretive operators,for which D(A) ⊆ K ⊆ ∩μ>0R(I + μA), D(B) ⊆ K ⊆ ∩μ>0R(I + μB) and A−1(0) ∩ B−1(0) �= ∅.Let {ηn}, {ϑn} and {ξn} be sequences of real numbers, for which 0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤

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ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1 for all n ∈ N. Let μ > 0, u1 ∈ K and PA−1(0)∩B−1(0)(u1) = u∗. Let{un} be defined by ⎧⎪⎪⎨⎪⎪⎩

wn = (1− ξn)un + ξn JAμ un,

zn = (1− ϑn)wn + ϑn JBμ wn,

un+1 = (1− ηn)JAμ wn + ηn JBμ zn, ∀ n ∈ N.

(16)

Then, we have the following:


(ii) limn→∞ ‖un − JAμ un‖ = 0 and limn→∞ ‖un − JBμ un‖ = 0.(iii){un} converges weakly to an element of A−1(0) ∩B−1(0) ∩ Bλ[u∗].

Proof. By assumption D(A) ⊆ K ⊆ ∩μ>0R(I + μA), we known that JAμ , JBμ : K → K benonexpansive. Note that D(A) ∩D(B) ⊆ K and hence

u∗ ∈ A−1(0) ∩B−1(0)⇒ u∗ ∈ D(A) ∩D(B) with 0 ∈ Au∗ and 0 ∈ Bu∗⇒ u∗ ∈ K with JAμ u∗ = u∗ and JBμ u∗ = u∗⇒ u∗ ∈ Fix(JAμ , JBμ ) ∩K.

Next, set S = JAμ and T = JBμ . Hence, Theorem 3 is the same way as Theorem 2.

4.2. Convexly Constrained Least Square Problem

We provide applications of Theorem 2 for finding solutions to common problems with twoconvexly constrained least square problems. We consider the following problem:

Let A,B ∈ B(H), and y, z ∈ H. Define ϕ, ψ : H→ R by

ϕ = ‖Au− y‖2 and ψ = ‖Bu− z‖2, ∀ u ∈ H,

where H is a real Hilbert space.Let K be a nonempty closed convex subset of H. The objective is to find b ∈ K such that

b ∈ arg minu∈K

ϕ(u) ∩ arg minu∈K

ψ(u), (17)

wherearg min

u∈Kϕ(u) := {u ∈ K : ϕ(u∗) = inf

u∈Kϕ(u)}.

Proposition 1 ([8]). Let H be a real Hilbert space, A ∈ B(H) with the adjoint A∗ and y ∈ H. Let K be anonempty closed convex subset of H. Let b ∈ H and δ ∈ (0, ∞). Then, the following statements are equivalent:

(i) b solves the following problem:minu∈K

‖Au− y‖2.

(ii) b = PK(b− δA∗(Ab− y)).(iii) 〈Av−Ab, y−Ab〉 ≤ 0, for all v ∈ K.

Theorem 4. Let K be a nonempty closed convex subset of a real Hilbert space H, y, z ∈ H and A,B ∈ B(H),for which the solution set of the problem in Equation (17) is nonempty. Let {ηn}, {ϑn} and {ξn} be sequencesof real numbers, for which 0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤ ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1 for all

427


n ∈ N. Let u1 ∈ H, Parg minu∈K ϕ(u)∩ arg minu∈K ψ(u)(u1) = u∗, δ ∈ (0, 2 min{ 1‖A‖2 , 1

‖B‖2 }), u1 ∈ K and{un} is defined by ⎧⎪⎪⎨⎪⎪⎩


zn = (1− ϑn)wn + ϑnTwn,

un+1 = (1− ηn)Swn + ηnTzn, ∀ n ∈ N.

(18)

where S,T : K → K defined by Su = PK(u − δA∗(Au − y)) and Tu = PK(u − δB∗(Bu − z)) for allu ∈ K. Then, we have the following:

(i) {un} is in the closed ball Bλ[u∗], where λ is a constant in (0, ∞) such that ‖u1 − u∗‖ ≤ λ.(ii) limn→∞ ‖un − Sun‖ = 0 and limn→∞ ‖un − Tun‖ = 0.(iii){un} converges weakly to an element of arg minu∈K ϕ(u) ∩ arg minu∈K ψ(u) ∩ Bλ[u∗].

Proof. Note that: ∇ϕ(u) = A∗(Au− y), for all u ∈ H; we obtain that ‖∇ϕ(u)−∇ϕ(v)‖ = ‖A∗(Au−y) − A∗(Av − y)‖ ≤ ‖A‖2‖u − v‖, for all u, v ∈ H. Thus, ∇ϕ is 1

‖A‖2 -ism and hence (I − δ∇ϕ) is

nonexpansive from K into H for σ ∈ (0, 2‖A‖2 ). Therefore, S = PK(I − σ∇ϕ) and T = PK(I − τ∇ϕ)

are nonexpansive mappings from K into itself for σ ∈ (0, 2‖A‖2 ) and τ ∈ (0, 2

‖B‖2 ), respectively. Hence,Theorem 4 is the same way as Theorem 2.

4.3. Convex Minimization Problem

We give an application to common solutions to convex programming problems in a Hilbert spaceH. We consider the following problem:

Let g1, g2 : H→ (−∞, ∞] be proper l.s.c. functions. The objective is to find x ∈ H such that:

x ∈ ∂g−11 (0) ∩ g−1

2 (0). (19)

Note that: J∂g1μ = proxμg1 .

Theorem 5. Let K be a nonempty closed convex subset of a real Hilbert space H. Let g1, g2 ∈ Γ0(H), forwhich the solution set of the problem in Equation (19) is nonempty. Let {ηn}, {ϑn} and {ξn} be sequences ofreal numbers, for which 0 < c1 ≤ ηn ≤ c1 < 1, 0 < c2 ≤ ϑn ≤ c2 < 1, 0 < c3 ≤ ξn ≤ c3 < 1 for all n ∈ N.Let μ > 0, u1 ∈ H and P

∂g−11 (0)∩g−1

2 (0)(u1) = u∗. Let u1 ∈ K and {un} is defined by

⎧⎪⎪⎨⎪⎪⎩wn = (1− ξn)un + ξn proxμg1(un),

zn = (1− ϑn)wn + ϑn proxμg2(wn),

un+1 = (1− ηn)proxμg1(wn) + ηn proxμg2(zn), ∀ n ∈ N.

(20)

Then, we have the following:

(i) {un} is in the closed ball Bλ[u∗], where λ is a constant in (0, ∞) such that ‖u1 − u∗‖ ≤ λ.(ii) limn→∞ ‖un − proxμg1(un)‖ = 0 and limn→∞ ‖un − proxμg2(un)‖ = 0.(iii){un} converges weakly to an element of ∂g−1

1 (0) ∩ g−12 (0) ∩ Bλ[u∗].

Proof. Using Lemma 1, we have that ∂g1 is maximal monotone. We know thatR(I + μ∂ f ) = H andusing the maximal monotonicity of ∂g1. Thus, J∂g1

μ = proxμg1 : H → H is nonexpansive. Similarly,

J∂g2μ = proxμg2 : H→ H is nonexpansive. Hence, Theorem 5 is the same way as Theorem 2.

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4.4. Signal Processing

We consider some applications of our algorithm to inverse problems occurring from signalprocessing. For example, we consider the following underdeterminated linear equation system:

y = Au + e, (21)

where u ∈ RN is recovered, y ∈ RM is observations or measured data with noisy e, and A : RN → RM

is a bounded linear observation operator. It determines a process with loss of information. For findingsolutions of the linear inverse problems in Equation (21), a successful one of some models is the convexunconstrained minimization problem:

minu∈RN

12‖Au− y‖2 + d‖u‖1, (22)

where d > 0 and ‖ · ‖1 is the l1−norm. Thus, we can find solution to Equation (22) by applying ourmethod in the case g1(u) = 1

2‖Au− y‖2 and g2(u) = d‖u‖1. For any α ∈ (0, 2L ], the corresponding

forward-backward operator Jg1,d‖·‖1α as follows:

Jg1,d‖·‖1α (u) = proxαd‖·‖1

(u− α∇g1(u)), (23)

where g1 is the squared loss function of the Lasso problem in Equation (22). The proximity operatorfor l1−norm is defined as the shrinkage operator as follows:

proxαd‖·‖1(u) = max(|ui| − αd, 0) · sgn(ui), (24)

where sgn(·) is the signum function. We apply the algorithm to the problem in Equation (22) follow asAlgorithm 2:

Algorithm 2: Three-step forward-backward operator

initialization: ηn, ϑn, ξn ∈ (0, 1), α, d ∈ (0, 1) u1 ∈ K and n = 1.while stopping criterion not met do

wn = (1− ξn)un + ξn Jg1,d‖·‖1α (un),

zn = (1− ϑn)wn + ϑn Jg1,d‖·‖1α (wn),

un+1 = (1− ηn)Jg1,d‖·‖1α (wn) + ηn Jg1,d‖·‖1

α (zn).end

In our experiment, we set the hits of a signal u ∈ RN . The matrix A ∈ RM×N was generatedfrom a normal distribution with mean zero and one invariance. The observation y is generated byGaussian noise distributed normally with mean 0 and variance 10−4. We compared our Algorithm 2with SPGA [12]. Let ηn = ϑn = ξn = 0.5, α = 0.1 and d = 0.01 in both Algorithm 2 and SPGA.The experiment was initialized by u1 = A∗y and terminated when ‖un+1−un‖

‖un‖ < 10−4. The restoration

accuracy was measured by means of the mean squared error: MSE = ‖u∗−u‖2

N , where u∗ is an estimatedsignal of u. All codes were written in Matlab 2016b and run on Dell i-5 Core laptop. We present thenumerical comparison of the results in Figures 1–6.

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Figure 1. From top to bottom: Original signal, observation data, recovered signal by Algorithm 2 andSPGA with N = 4096, M = 1024 and 10 spikes, respectively.

Figure 2. Comparison MSE of two algorithms for recovered signal with N = 4096, M = 1024 and10 spikes, respectively.

430




431




5. Conclusions

In this work, we introduce a modified iterative scheme in Banach spaces and solve common zerosof accretive operators, convexly constrained least square problem, convex minimization problem andsignal processing. In the case of signal processing, all results are compared with the forward-backwardmethod in Algorithm 2 and SPGA, as proposed in [12]. The numerical results show that Algorithm 2has a better convergence behavior than SPGA when using the same step sizes for both.

Author Contributions: A.P. and P.S.; writing original draft, A.P. and P.S.; data analysis, A.P. and P.S.; formalanalysis and methodology

Funding: This research was funded by Rajamangala University of Technology Thanyaburi (RMUTT).

Acknowledgments: The first author thanks Rambhai Barni Rajabhat University for the support. PakeetaSukprasert was financially supported by Rajamangala University of Technology Thanyaburi (RMUTT).

432



Abbreviations

The following abbreviations are used in this manuscript:

Symbols Display

l.s.c. lower semicontinuous, convexB(H) the set of all bounded and linear operators from H into itselfr.u.c. real uniformly convex

References

1. Kankam, K.; Pholasa, N.; Cholamjiak, P. On convergence and complexity of the modified forward-backwardmethod involving new linesearches for convex minimization. Math. Meth. Appl. Sci. 2019, 42, 1352–1362.[CrossRef]

2. Candès, E.J.; Wakin, M.B. An introduction to compressive sampling. IEEE Signal Process. Mag. 2008, 25, 21–30.[CrossRef]

3. Suantai, S.; Kesornprom, S.; Cholamjiak, P. A new hybrid CQ algorithm for the split feasibilityproblem in Hilbert spaces and Its applications to compressed Sensing. Mathematics 2019, 7, 789;doi:10.3390/math7090789. [CrossRef]

4. Kitkuan, D.; Kumam, P.; Padcharoen, A.; Kumam, W.; Thounthong, P. Algorithms for zeros of two accretiveoperators for solving convex minimization problems and its application to image restoration problems.J. Comput. Appl. Math. 2019, 354, 471–495. [CrossRef]

5. Padcharoen, A.; Kumam, P.; Cho, Y.J. Split common fixed point problems for demicontractive operators.Numer. Algorithms 2019, 82, 297–320. [CrossRef]

6. Cholamjiak, P.; Shehu, Y. Inertial forward-backward splitting method in Banach spaces with application tocompressed sensing. Appl. Math. 2019, 64, 409–435. [CrossRef]

7. Jirakitpuwapat, W.; Kumam, P.; Cho, Y.J.; Sitthithakerngkiet, K. A general algorithm for the split commonfixed point problem with its applications to signal processing. Mathematics 2019, 7, 226. [CrossRef]

8. Combettes, P.L.; Wajs, V.R. Signal recovery by proximal forward-backward splitting. Multiscale Model Simul.2005, 4, 1168–1200. [CrossRef]

9. Picard, E. Memoire sur la theorie des equations aux d’erives partielles et la methode des approximationssuccessives. J. Math Pures Appl. 1890, 231, 145–210.

10. Mann, W.R. Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. [CrossRef]11. Ishikawa, S. Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44, 147–150. [CrossRef]12. Agarwal, R.P.; O’Regan, D.; Sahu, D.R. Iterative construction of fixed points of nearly asymptotically

nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8, 61–79.13. Sahu, V.K.; Pathak, H.K.; Tiwari, R. Convergence theorems for new iteration scheme and comparison results.

Aligarh Bull. Math. 2016, 35, 19–42.14. Thakur, B.S.; Thakur, D.; Postolache, M. New iteration scheme for approximating fixed point of

non-expansive mappings. Filomat 2016, 30, 2711–2720. [CrossRef]15. Chang, S.S.; Wen, C.F.; Yao, J.C. Zero point problem of accretive operators in Banach spaces. Bull. Malays.

Math. Sci. Soc. 2019, 42, 105–118. [CrossRef]16. Browder, F.E. Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math.

Soc. 1967, 73, 875–882. [CrossRef]17. Browder, F.E. Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc.

1968, 7, 660–665. [CrossRef]18. Cioranescu, I. Geometry of Banach Spaces, Duality Mapping and Nonlinear Problems; Kluwer:

Amsterdam, The Netherlands, 1990.19. Takahashi, W. Nonlinear Functional Analysis, Fixed Point Theory and Its Applications; Yokohama Publishers:

Yokohama, Japan, 2000.20. Rockafellar, R.T. On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 1970, 33, 209–216.

[CrossRef]

433


21. Xu, H.K. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16, 1127–1138. [CrossRef]22. Opial, Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings.

Bull. Am. Math. Soc. 1967, 73, 591–597. [CrossRef]23. Sahu, D.R.; Pitea, A.; Verma, M. A new iteration technique for nonlinear operators as concerns convex

programming and feasibility problems. Numer. Algorithms 2019. [CrossRef]24. Goebel, K.; Reich, S. Uniform Convexity, Hyperbolic Geometry and Non Expansive Mappings; Marcel Dekker:

New York, NY, USA; Basel, Switzerland, 1984 .


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mathematics

Article

Hybrid Second Order Method for OrthogonalProjection onto Parametric Curve in n-DimensionalEuclidean Space

Juan Liang 1,2,†, Linke Hou 3,†,*, Xiaowu Li 4,†,* , Feng Pan 4,†, Taixia Cheng 5,† and Lin Wang 4,†

1 Data Science and Technology, North University of China, Taiyuan 030051, Shanxi, China;[email protected]

2 Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, Shanxi, China3 Center for Economic Research, Shandong University, Jinan 250100, Shandong, China4 College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou,

China; [email protected] (F.P.); [email protected] (L.W.)5 Graduate School, Guizhou Minzu University, Guiyang 550025, Guizhou, China; [email protected]* Correspondence: [email protected] (L.H.); [email protected] (X.L.);

Tel.: +86-135-0640-1186 (L.H.); +86-187-8613-2431 (X.L.)† These authors contributed equally to this work.

Received: 16 October 2018; Accepted: 28 November 2018; Published: 5 December 2018

Abstract: Orthogonal projection a point onto a parametric curve, three classic first order algorithmshave been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter,H-H-H method). In this research, we give a proof of the approach’s first order convergence and itsnon-dependence on the initial value. For some special cases of divergence for the H-H-H method,we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybridsecond order method for orthogonal projection onto parametric curve in an n-dimensional Euclideanspace (hereafter, our method). Our method essentially utilizes hybrid iteration, so it convergesfaster than current methods with a second order convergence and remains independent from theinitial value. We provide some numerical examples to confirm robustness and high efficiency ofthe method.

Keywords: point projection; intersection; parametric curve; n-dimensional Euclidean space; Newton’ssecond order method; fixed point theorem

1. Introduction

In this research, we will discuss the minimum distance problem between a point and a parametriccurve in an n-dimensional Euclidean space, and how to gain the closest point (footpoint) on the curveas well as its corresponding parameter, which is termed as the point projection or inversion problem ofa parametric curve in an n-dimensional Euclidean space. It is an important issue in the themes such asgeometric modeling, computer graphics, computer-aided geometry design (CAGD) and computervision [1,2]. Both projection and inversion are fundamental for a series of techniques, for instance, theinteractive selection of curves and surfaces [1,3], the curve fitting [1,3], reconstructing curves [2,4,5]and projecting a space curve onto a surface [6]. This vital technique is also used in the ICP (iterativeclosest point) method for shape registration [7].

The Newton-Raphson algorithm is deemed as the most classic one for orthogonal projectiononto parametric curve and surface. Searching the root of a polynomial by a Newton-Raphsonalgorithm can be found in [8–12]. In order to solve the adaptive smoothing for the standard finiteunconstrained minimax problems, Polak et al. [13] have presented a extended Newton’s algorithmwhere a new feedback precision-adjustment rule is used in their extended Newton’s algorithm.



Once the Newton-Raphson method reaches its convergence, two advantages emerge and it convergesvery fast with high precision. However, the result relies heavily on a good guess of initial value in theneighborhood of the solution.

Meanwhile, the classic subdivision method consists of several procedures: Firstly, subdivideNURBS curve or surface into a set of Bézier sub-curves or patches and eliminate redundancy orunnecessary Bézier sub-curves or Bézier patches. Then, get the approximation candidate points. Finally,get the closest point through comparing the distances between the test point and candidate points.This technique is reflected in [1]. Using new exclusion criteria within the subdivision strategy,the robustness for the projection of points on NURBS curves and surfaces in [14] has beenimproved than that in [1], but this criterion is sometimes too critical. Zou et al. [15] use subdivisionminimization techniques which rely on the convex hull characteristic of the Bernstein basis toimpute the minimum distance between two point sets. They transform the problem into solvingof n-dimensional nonlinear equations, where n variables could be represented as the tensor productBernstein basis. Cohen et al. [16] develop a framework for implementing general successive subdivisionschemes for nonuniform B-splines to generate the new vertices and the new knot vectors which aresatisfied with derived polygon. Piegl et al. [17] repeatedly subdivide a NURBS surface into fourquadrilateral patches and then project the test point onto the closest quadrilateral until it can findthe parameter from the closest quadrilateral. Using multivariate rational functions, Elber et al. [11]construct a solver for a set of geometric constraints represented by inequalities. When the dimensionof the solver is greater than zero, they subdivide the multivariate function(s) so as to bind the functionvalues within a specified domain. Derived from [11] but with more efficiency, a hybrid parallelmethod in [18] exploits both the CPU and the GPU multi-core architectures to solve systems undermultivariate constraints. Those GPU-based subdivision methods essentially exploit the parallelisminherent in the subdivision of multivariate polynomial. This geometric-based algorithm improvesin performance compared to the existing subdivision-based CPU. Two blending schemes in [19]efficiently remove no-root domains, and hence greatly reduce the number of subdivisions. Through asimple linear combination of functions for a given system of nonlinear equations, no-root domainand searching out all control points for its Bernstein-Bézier basic with the same sign must be satisfiedwith the seek function. During the subdivision process, it can continuously create these kinds offunctions to get rid of the no-root domain. As a result, van Sosin et al. [20] efficiently form variouscomplex piecewise polynomial systems with zero or inequality constraints in zero-dimensional orone-dimensional solution spaces. Based on their own works [11,20], Barton et al. [21] propose a newsolver to solve a non-constrained (piecewise) polynomial system. Two termination criteria are appliedin the subdivision-based solver: the no-loop test and the single-component test. Once two terminationcriteria are satisfied, it then can get the domains which have a single monotone univariate solution.The advantage of these methods is that they can find all solutions, while their disadvantage is thatthey are computationally expensive and may need many subdivision steps.

The third classic methods for orthogonal projection onto parametric curve and surfaceare geometry methods. They are mainly classified into eight different types of geometrymethods: tangent method [22,23], torus patch approximating method [24], circular or sphericalclipping method [25,26], culling technique [27], root-finding problem with Bézier clipping [28,29],curvature information method [6,30], repeated knot insertion method [31] and hybrid geometrymethod [32]. Johnson et al. [22] use tangent cones to search for regions with satisfaction of distanceextrema conditions and then to solve the minimum distance between a point and a curve, but itis not easy to construct tangent cones at any time. A torus patch approximatively approaches forpoint projection on surfaces in [24]. For the pure geometry method of a torus patch, it is difficult toachieve high precision of the final iterative parametric value. A circular clipping method can removethe curve parts outside a circle with the test point being the circle’s center, and the radius of theelimination circle will shrink until it satisfies the criteria to terminate [26]. Similar to the algorithm [26],a spherical clipping technique for computing the minimum distance with clamped B-spline surface

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is provided by [25]. A culling technique to remove superfluous curves and surfaces containing noprojection from the given point is proposed in [27], which is in line with the idea in [1]. Using Newton’smethod for the last step [1,25–27], the special case of non-convergence may happen. In view of theconvex-hull property of Bernstein-Bézier representations, the problem to be solved can be formulatedas a univariate root-finding problem. Given a C1 parametric curve c(t) and a point p, the projectionconstraint problem can be formulated as a univariate root-finding problem 〈c′(t), c(t)− p〉 = 0 with ametric induced by the Euclidean scalar product in Rn. If the curve is parametrized by a (piece-wise)polynomial, then the fast root-finding schemes as a Bézier clipping [28,29] can be used. The only issueis the C1 discontinuities that can be checked in a post-process. One advantage of these methods is thatthey do not need any initial guess on the parameter value. They adopt the key technology of degreereduction via clipping to yield a strip bounded of two quadratic polynomials. Curvature informationis found for computing the minimum distance between a point and a parameter curve or surfacein [6,30]. However, it needs to consider the second order derivative and the method [30] is not fit forn-dimensional Euclidean space. Hu et al. [6] have not proved the convergence of their two algorithms.Li et al. [33] have strictly proved convergence analysis for orthogonal projection onto planar parametriccurve in [6]. Based on repeated knot insertion, Mørken et al. [31] exploit the relationship between aspline and its control polygon and then present a simple and efficient method to compute zeros ofspline functions. Li et al. [32] present the hybrid second order algorithm which orthogonally projectsonto parametric surface; it actually utilizes the composite technology and hence converges nicely withconvergence order being 2. The geometric method can not only solve the problem of point orthogonalprojecting onto parametric curve and surface but also compute the minimum distance betweenparametric curves and parametric surfaces. Li et al. [23] have used the tangent method to computethe intersection between two spatial curves. Based on the methods in [34,35], they have extended tocompute the Hausdorff distance between two B-spline curves. Based on matching a surface patchfrom one model to the other model which is the corresponding nearby surface patch, an algorithmfor solving the Hausdorff distance between two freeform surfaces is presented in Kim et al. [36],where a hierarchy of Coons patches and bilinear surfaces that approximate the NURBS surfaceswith bounding volume is adopted. Of course, the common feature of geometric methods is that theultimate solution accuracy is not very high. To sum up, these algorithms have been proposed to exploitdiverse techniques such as Newton’s iterative method, solving polynomial equation roots methods,subdividing methods, geometry methods. A review of previous algorithms on point projection andinversion problem is obtained in [37].

More specifically, using the tangent line or tangent plane with first order geometric information,a classical simple and efficient first order algorithm which orthogonally project onto parametric curveand surface is proposed in [38–40] (H-H-H method). However, the proof of the convergence for theH-H-H method can not be found in this literature. In this research, we try to give two contributions.Firstly, we give proof that the algorithm is first order convergent and it does not depend on theinitial value. We then provide some numerical examples to show its high convergence rate. Secondly,for several special cases where the H-H-H method is not convergent, there are two methods (Newton’smethod and the H-H-H method) to combine our method. If the H-H-H method’s iterative parametricvalue is satisfied with the convergence condition of the Newton’s method, we then go to Newton’smethod to increase the convergence process. Otherwise, we go on the H-H-H method until its iterativeparametric value is satisfied with the convergence condition of the Newton’s method, and we thenturn to it as above. This algorithm not only ensures the robustness of convergence, but also improvesthe convergence rate. Our hybrid method can go faster than the existing methods and ensures theindependence to the initial value. Some numerical examples verify our conclusion.

The rest of this paper is arranged as follows. In Section 2, convergence analysis of the H-H-Hmethod is presented. In Section 3, for several special cases where the H-H-H method is not convergent,an improved our method is provided. Convergence analysis for our method is also provided in this

437


section. In Section 4, some numerical examples for our method are verified. In Section 5, conclusions areprovided.

2. Convergence Analysis of the H-H-H Method

In this part, we will prove that the algorithm defined by Equations (2) or (3) is of firstorder convergence and its convergence does not rely on the initial value. Suppose a C2 curvec(t) = ( f1(t), f2(t), . . . , fn(t)) in an n-dimensional Euclidean space Rn(n ≥ 2) and a test pointp = (p1, p2, . . . , pn). The first order geometric method to compute the footpoint q of test point pcan be implemented as below. Projecting test point p onto the tangent line of the parametric curve c(t)in an n-dimensional Euclidean space at t = tm gets a point q determined by c(tm) and its derivativec′(tm). The footpoint can be approximated as

q = c(tm) + Δtc′(tm). (1)

Then,

Δt =〈c′(tm), p− c(tm)〉〈c′(tm), c′(tm)〉 , (2)

where 〈x, y〉 is the scalar product of vectors x, y ∈ Rn. Equation (2) can also be expressed as

K1(tm) = tm +〈c′(tm), p− c(tm)〉〈c′(tm), c′(tm)〉 . (3)

Let tm ← K1(tm), and repeatedly iterate the above process until |K1(tm)− tm| is less than an errortolerance ε. This method is addressed as H-H-H method [38–40]. Furthermore, convergence of thismethod will not depend on the choice of the initial value. According to many of our test experiments,when the iterative parametric value approaches the target parametric value α, the iteration step sizebecomes smaller and smaller, while the corresponding number of iterations becomes bigger and bigger.

Theorem 1. The convergence order of the method defined by Equations (2) or (3) is one, and itsconvergence does not depend on the initial value.

Proof. We adopt the numerical analysis method which is equivalent to those in the literature [41,42].Firstly, we deduce the expression of footpoint q. Suppose that parameter curve c(t) is a C2 curve in ann-dimensional Euclidean space Rn(n ≥ 2), where the corresponding projecting point with parameter α

is orthogonal projecting of the test point p = (p1, p2, . . . , pn) onto the parametric curve c(t). It is easyto indicate a relational expression

〈p− h, n〉 = 0, (4)

where h = c(α) and tangent vector n = c′(α). In order to solve the intersection (footpoint q) betweenthe tangent line, which goes through the parametric curve c(t) at t = tm, and the perpendicular line,which is determined by the test point p, we try to express the equation of the tangent line as:

x = c(tm) + c′(tm) · s, (5)

where x = (x1, x2, . . . , xn) and s is a parameter. In addition, the vector of line segment both goingthrough the test point p and the point c(tm) will be

y = p− x, (6)

where y = (y1, y2, . . . , yn). Because the vector (6) and the tangent vector c′(tm) of Equation (5) areorthogonal to each other, the current parameter value s of Equation (5) is

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s0 =〈p− c(tm), c′(tm)〉〈c′(tm), c′(tm)〉 . (7)

Substituting (7) into (5), we have

q = c(tm) + c′(tm) · s0. (8)

Thus, the footpoint q = (q1, q2, . . . , qn) is determined by Equation (8).Secondly, we deduce that the convergence order of the method defined by (2) or (3) is first order

convergent. Our proof method absorbs the idea of [41,42]. Substituting (8) into (2), and simplifying,we get the relationship,

Δt =〈p− c(tm), c′(tm)〉〈c′(tm), c′(tm)〉 . (9)

Using Taylor’s expansion, we get

c(tm) = B0 + B1em + B2e2m + o(e3

m), (10)

c′(tm) = B1 + 2B2em + o(e2m), (11)

where em = tm − α, and Bi = (1/i!)c(i)(α), i = 0, 1, 2, . . . From (10) and (11) and combining with (4),the numerator of Equation (9) can be transformed into the following one:

〈p− c(tm), c′(tm)〉= L1em + L2e2

m + o(e3m),

(12)

where L1 = 2 〈p− B0, B2〉 − 〈B1, B1〉 , L2 = −3 〈B1, B2〉. By (11), the denominator of Equation (9) canbe changed as follows:

〈c′(tm), c′(tm)〉= M1 + M2em + M3e2

m + o(e3m),

(13)

where M1 = 〈B1,B1〉 , M2 = 4 〈B1,B2〉 , M3 = 4 〈B2,B2〉. Substituting Equations (12) and (13) into theright-hand side of Equation (9), we get

Δt =〈p− c(tm), c′(tm)〉〈c′(tm), c′(tm)〉

=L1em + L2e2

m + o(e3m)

M1 + M2em + M3e2m + o(e3

m).

(14)

Using Taylor’s expansion by Maple 18, and through simplification, we get

K1(tm) =α + (L1

M1+ 1)em +

L2M1 − L1M2

M21

e2m + o(e3

m),

=α + (L1

M1+ 1)em + o(e2

m),

=α + C0em + o(e2m),

(15)

where the symbol C0 is the coefficient of the first order error em of Equation (15). The result implies theiterative Equations (2) or (3) is of first order convergence.

Now, we try to interpret that Equations (2) or (3) do not depend on the initial value.Our proof method absorbs the idea of references [43,44]. Without loss of generality, we only prove

that convergence of Equations (2) or (3) does not depend on the initial value in two-dimensional case.As to convergence of Equations (2) or (3) not being dependent on the initial value in generaln-dimensional Euclidean space case, it is completely equivalent to the two-dimensional case.

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Firstly, we interpret Figure 1. For a horizontal axis t, there are two points are on the planarparametric curve c(t). For the first point c(tm) on the horizontal axis, the test point p orthogonalprojects it onto the planar parametric curve c(t) and yields the second point and its correspondingparameter value α on the horizontal axis. Then, by the iterative methods (2) or (3), the line segmentconnected by the point p and the point c(α) is perpendicular to the tangent line of the planar parametriccurve c(t) at t = α. The footpoint q is determined by the tangent line of the planar parametric curvec(t) through the point c(tm). Evidently, the parametric value tm+1 of footpoint q can be used as thenext iterative value. M is the corresponding parametric value of the middle point of the point c(tm)

and the footpoint q.

c( )

c(t )m

tm

c(t)

tm+1

Figure 1. Geometric illustration for convergence analysis.

Secondly, we prove the argument whose convergence of Equations (2) or (3) does not depend onthe initial value. It is easy to know that t denotes the corresponding parameter for the first dimensionalof the planar parametric curve on the two-dimensional plane. When the iterative Equations (2)or (3) start to run, we suppose that the iterative parameter value is satisfied with the inequalityrelationship tm < α and the corresponding parameter of the footpoint q is tm+1, as shown in Figure 1.

The middle point of two points (tm+1, 0) and (tm, 0) is (M, 0), i.e., M =tm + tm+1

2, and, because of

0 < Δt = tm+1 − tm, then there exists an inequality tm < M < α. Equivalently, tm − α < tm+1 − α <

α − tm = −(tm − α), which can be expressed as |em+1| < |em|, where em = tm − α. If tm > α, wecan get the same result through the same method. Thus, an iterative error expression |em+1| < |em|in a two-dimensional plane is demonstrated. Thus, it is known that convergence of the iterativeEquations (2) or (3) does not depend on the initial value in two-dimensional planes (see Figure 1).Furthermore, we could get the argument that convergence of the iterative Equations (2) or (3) does notdepend on the initial value in an n-dimensional Euclidean space. The proof is completed.

3. The Improved Algorithm

3.1. Counterexamples

In Section 2, convergence of the H-H-H method does not depend on the initial value. For specialcases with non-convergence by the H-H-H method, we then enumerate nine counterexamples.

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Counterexample 1. There are a parametric curve c(t) = (t, 1 + t2) and a test point p = (0, 0).The projection point and parametric value of the test point p are (0, 1) and α = 0, respectively.As to many initial values, the H-H-H method fails to converge to α. When the initial values are t= −3,−2,−1.5, 1.5, 2, 3, respectively, there repeatedly appear alternating oscillatory iteration values of0.412415429665, −0.412415429665. Furthermore, for a parametric curve c(t) = (t, 1 + a1t2 + a2t4 + a3t6 +

a4t8 + a5t10), a1 �= 0, a2 �= 0, a3 �= 0, a4 �= 0, a5 �= 0, about p = (0, 0) and many initial values, the H-H-Hmethod fails to converge to α (see Figure 2).

Figure 2. Geometric illustration for counterexample 1.

Counterexample 2. There are a parametric curve c(t) = (t, t2, t4, t6, 1 + t2 + t4 + t6 + t8) anda test point p = (0, 0, 0, 0, 0). The projection point and parametric value of the test point p are(0, 0, 0, 0, 1) and α = 0, respectively. For any initial value, the H-H-H method fails to converge toα. When the initial values are t = −5,−4,−3,−2,−1, 1, 2, 3, 4, 5, respectively, there repeatedlyappear alternating oscillatory iteration values of 0.304949569175, −0.304949569175. Furthermore,for a parametric curve c(t) = (a0t, a1t2,a2t4,a3t6,1 + a4t2 + a5t4 + a6t6 + a7t8 + a8t10 + a9t28),a0 �= 0, a1 �= 0, a2 �= 0, a3 �= 0, a4 �= 0, a5 �= 0, a6 �= 0, a7 �= 0, a8 �= 0, a9 �= 0, about point p = (0, 0, 0, 0, 0)and any initial value, the H-H-H method fails to converge to α.

Counterexample 3. There are a parametric curve c(t) = (t, sin(t)), t ∈ [0, 3] and a test pointp = (4, 9). The projection point and parametric value of the test point p are (1.842576, 0.9632946) andα = 1.842576, respectively. For point p and any initial value, the H-H-H method fails to converge to α.When the initial values are t = −5,−4,−3,−2,−1, 1, 2, 3, 4, 5, respectively, there repeatedly appear alternatingoscillatory iteration values of 2.165320, 0.0778704, 6.505971, 9.609789. In addition, for a parametric curvec(t) = (t, sin(at)), a �= 0, for any test point p and any initial value, the H-H-H method fails to converge to α

(see Figure 3).

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Figure 3. Geometric illustration of counterexample 3.

Counterexample 4. There are a parametric curve c(t) = (t, cos(t)), t ∈ [0, 3] and a test point p = (2, 6).The projection point and parametric value of the test point p are (0.3354892, 0.9442493) and α = 0.3354892,respectively. For test point p and any initial value, the H-H-H method fails to converge to α. When the initialvalue is t = −5, alternating oscillatory iteration values of 5.18741299662, 3.59425803253, −0.507188248308,1.6901041247, 3.82746208506 repeatedly appear. When the initial value is t = 2, very irregular oscillatoryiteration values of 0.652526561595, −0.720371663877, −2.39555359952, 0.365881194752, 2.06880954777,3.18725085474, 1.71447110647, etc. appear In addition, for a parametric curve c(t) = (t, cos(at)), a �= 0,for any test point p and any initial value, the H-H-H method fails to converge to α (see Figure 4).

Figure 4. Geometric illustration of counterexample 4.

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Counterexample 5. There are a parametric curve c(t) = (t, t, t, t, sin(t)), t ∈ [6, 9] and a test pointp = (3, 5, 7, 9, 11). The projection point and parametric value of the test point p are (7.310786, 7.310786,7.310786, 7.310786, 0.8560612) and α = 7.310786, respectively. For point p and any initial value, the H-H-Hmethod fails to converge to α. When the initial values are t = −9,−7,−5, 6, 8, respectively, there repeatedlyappear alternating oscillatory iteration values of 7.24999006346, 6.37363460615. In addition, for a parametriccurve c(t) = (t, t, t, t, sin(at)), t ∈ [6, 9], a �= 0 with a test point p = (3, 5, 7, 9, 11), for any initial value,the H-H-H method fails to converge to α.

Counterexample 6. There are a parametric curve c(t) = (t, t, t, t, cos(t)), t ∈ [4, 8] and a testpoint p = (2, 4, 6, 8, 10). The projection point and parametric value of the test point p are (5.883406,5.883406, 5.883406, 5.883406, 0.9211469) and α = 5.883406, respectively. For point p and any initial value,the H-H-H method fails to converge to α. When the initial values are t =,−4,−3,−2, 4, 5, 6, 7, respectively,there repeatedly appear alternating oscillatory iteration values of 4.17182145828, 7.80116702003. In addition,about a parametric curve c(t) = (t, t, t, t, cos(at)), t ∈ [4, 8], a �= 0 with a point p = (2, 4, 6, 8, 10), for anyinitial value, the H-H-H method fails to converge. The non-convergence explanation of the three counterexamplesbelow are similar to the preceding six ones and omitted to save space.

Counterexample 7. There are a parametric curve c(t) = (t4 + 2t2 + 1, t2 + 1, t4 + 2, t2, 3t6 + t4 + 2t2)

in five-dimensional Euclidean space and a test point p = (0, 0, 0, 0, 0). The projection point and parametricvalue of the test point p are (1, 1, 2, 0, 0) and α = 0, respectively. For any initial value t0, the H-H-H methodfails to converge. We also test many other examples, such as when parametric curve is completely symmetricaland the point is on the symmetrical axis of parametric curve. For any initial value t0, the same results remain.

Counterexample 8. There are a parametric curve c(t) = (t,sin(t), t,sin(t),sin(t)), t ∈ [−5, 5] infive-dimensional Euclidean space and a test point p = (3, 4, 5, 6, 7). The corresponding orthogonal projectionparametric value α are −3.493548, −2.280571, 1.875969, 4.791677, respectively. For any initial value t0,the H-H-H method fails to converge.

Counterexample 9. There is a parametric curve c(t) =(sin(t),cos(t), t, sin(t),cos(t)), t ∈ [−5, 5]in five-dimensional Euclidean space and a test point p = (3, 4, 5, 6, 7). The corresponding orthogonal projectionparametric value α are −4.833375, −3.058735, 0.9730030, 3.738442, respectively. For any initial value t0,the H-H-H method fails to converge.

3.2. The Improved Algorithm

Due to the H-H-H method’s non-convergence for some special cases, the improved algorithmis presented to ensure the converge for any parametric curve, test point and initial value. The mostclassic Newton’s method can be expressed as

tm+1 = tm − f (tm)

f ′(tm), (16)

where f (t) =< T1, V1 >= 0, T1 = c′(t), V1 = p− c(t). It converges faster than the H-H-H method.However, the convergence of this depends on the chosen initial value. Only when the local convergencecondition for the Newton’s method is satisfied, the method can acquire high effectiveness. In orderto improve the robustness and rate of convergence, based on the the H-H-H method, our method isproposed. Combining the respective advantage of their two methods, if the iterative parametric valueof the H-H-H method is satisfied with the convergence condition of the Newton’s method, we then goto the method to increase the convergence process. Otherwise, we continue the H-H-H method until itcan generate iterative parametric value while satisfying the convergence condition by the Newton’smethod, and we then go to the iterative process mentioned above. Thus, we run to the end of thewhole process. The procedure not only ensures the robustness of convergence, but also improves the

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convergence rate. Using a hybrid strategy, our method is faster than current methods and independentfrom the initial value. Some numerical examples verify our conclusion. Our method can be realized asfollows (see Figure 5).

c t( ) tm + 1

tmtm

p

q

(a)0-1 1

K (t )1 m

( )T

(b)

tm

tm+1

tm+2

v=f(t)

t

v

α

(c)

Figure 5. Geometric illustration for our method. (a) Running the H-H-H method; (b) Judging theH-H-H method whether being satisfied the convergence condition of fixed point theorem for theNewton’s iterative method; (c) Running the Newton’s iterative method.

Hybrid second order method

Input: Initial iterative value t0, test point p and parametric curve c(t) in an n-dimensionalEuclidean space.Output: The corresponding parameter α determined by orthogonal projection point.Step 1. Initial iterative parametric value t0 is input.Step 2. Using the iterative Equation (3), calculate the parametric value K1(t0), and update K1(t0) to t1,

namely, t1 = K1(t0).Step 3. Determine whether absolute value of difference between the current t0 and the new t1 is near 0.

If so, this algorithm is ended.Step 4. Substitute the new t1 into

∣∣∣ f (t) f ′′(t)f ′(t)2

∣∣∣, determine if∣∣∣ f (t1) f ′′(t1)

f ′(t1)2

∣∣∣ < 1.

If (∣∣∣ f (t1) f ′′(t1)

f ′(t1)2

∣∣∣ < 1) {

Using Newton’s iterative Equation (16), compute t0 = t1 − f (t1)f ′(t1)

until absolute value ofdifference between the current t1 and the new t0 is near 0; then, this algorithm ends.}Else {

turn to Step 2.

}

Remark 1. Firstly, a geometric illustration of our method in Figure 5 would be presented. Figure 5a illustratesthe second step of our method where the next iterative parameter value tm+1 = K1(tm) = tm + 〈c′(tm),p−c(tm)〉

〈c′(tm),c′(tm)〉is determined by the iterative Equation (3). During the iterative process, the step Δt will become smaller andsmaller. Thus, the next iterative parameter value tm+1 comes close to parameter value tm but far from thefootpoint q. If the third step of our method is not over, then our method goes into the fourth step. Figure 5bis judging condition of a fixed point theorem of the fourth step of our method. If T =

∣∣∣ f (t) f ′′(t)f ′(t)2

∣∣∣ < 1, then itturns to the Newton’s method in Figure 5c until it runs to the end of the whole process of Newton’s second orderiteration; otherwise, it goes to the second step in Figure 5a.

Secondly, we give an interpretation for the singularity case of the iterative Equation (16). As to somespecial cases where the H-H-H method is not convergent in Section 3.1, our method still converges. We testmany examples for arbitrary initial value, arbitrary test point and arbitrary parametric curve and find that ourmethod remains more robust to converge than the H-H-H method. If the first order derivative f ′(tm) of theiterative Equation (16) develops into 0, i.e., f ′(tm) = 0 about some non-negative integer m, we use a perturbedmethod to solve the special problem, which adopts the idea in [23,45]. Namely, the function f ′(tm) = 0 could be

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increased by a very small positive number ε, i.e., f ′(tm) = f ′(tm) + ε, and then the iteration by Equation (16)is continued in order to calculate the parameter value. On the other hand, if the curve can be parametrized bya (piece-wise) polynomial, then the fast root-finding schemes such as Bézier clipping [28,29] are efficient ones.The only issue is the C1 discontinuities that can be checked in a post-process. One then does not need any initialguess on the parameter value.

Thirdly, if the curve is only C0 continuous, and the closest point can be exactly such a point, then thederivative is not well defined and our method may fail to find such a point. Namely, there are singular pointson the parametric curve. We adopt the following technique to solve the problem of singularity. We use themethods [46–48] to find all singular points on the parametric curve and the corresponding parametric value ofeach singular point as many as possible. Then, the hybrid second order method comes into work. If the currentiterative parametric value tm is the corresponding parametric value of a singular point, we make a very smallperturbation ε to the current iterative parametric value tm, i.e., tm = tm + ε. The purpose of this behavior is toenable the hybrid second order method to run normally. Then, from all candidate points (singular points andorthogonal projection points), a corresponding point is selected so that the distance between the correspondingpoint and the test point is the minimum one. When the entire program terminates, the minimum distance and itscorresponding parameter value are found.

3.3. Convergence Analysis of the Improved Algorithm

In this subsection, we prove the convergence analysis of our method.

Theorem 2. In Reference [49] (Fixed Point Theorem)

If φ(x) ∈ C[c, d], φ(x) ∈ [c, d] for all x ∈ [c, d]; furthermore, if φ′(x) exists on (c, d) and a positiveconstant L < 1 exists with |φ′(x)| ≤ L for all x ∈ (c, d), then there exists exactly one fixed point in [c, d].

In addition, if φ(t) = t− f (t)f ′(t) , the corresponding fixed point theorem of Newton’s method is

as follows:

Theorem 3. Let f : [c, d]→ [c, d] be a differentiable function, if for all t ∈ [c, d], there is∣∣∣∣ f (t) f ′′(t)f ′2(t)

∣∣∣∣ < 1. (17)

Then, there is a fixed point l0 ∈ [c, d] in Newton’s iteration expression (16) such that

l0 = l0 − f (l0)f ′(l0)

. Meanwhile, the iteration sequence {tm} been from expression (16) can converge to

the fixed point when ∀t0 ∈ [c, d].

Theorem 4. Our method is second order convergent.

Proof: Let α be a simple zero for a nonlinear function f (t) =< T1, V1 >= 0, where T1 = c′(t), V1 =

p− c(t). Using Taylor’s expansion, we have

f (tm) = f ′(α)[em + b2e2m + b3e3

m + o(e4m)], (18)

f ′(tm) = f ′(α)[2b2em + 3b3e2m + o(e3

m)], (19)

where bk =f (k)(α)k! f ′(α) , k = 2, 3, . . . , and em = tm − α. Combining with (15), we then have

ym = φ(tm) = tm − f (tm)

f ′(tm)= α + b2C2

0e2m + o(e3

m). (20)

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This means that the convergence order of our method is 2. The proof is completed. �

Theorem 5. Convergence of our method does not depend on the initial value.

Proof. According to the description of our method, if the iterative parametric value of the H-H-Hmethod is satisfied with the convergence condition of the Newton’s method, we then go to theNewton’s method. Otherwise, we steadily adopt the H-H-H method until its iterative parametricvalue is satisfied with the convergence condition of the Newton’s method, and we go to Newton’smethod. Then, we run to the end of the whole process. Theorem 1 ensures that it does not depend onthe initial value. If our method goes to the fourth step and if it is appropriate to the condition of thefixed point theorem (Theorem 3), Newton’s method is realized by our method. Then, the fourth stepof our method being also independent of the initial value can be confirmed by Theorem 3. In brief,convergence of our method does not depend on the initial value via the whole algorithm executionprocess. The proof is completed.


In order to illustrate the superiority of our method to other algorithms, we provide five numericalexamples to confirm its robustness and high efficiency. From Tables 1–14, the iterative terminationcriteria is satisfied such that |tm − α| < 10−17and |tm+1 − tn| < 10−17. All numerical results werecomputed through g++ in a Fedora Linux 8 environment. The approximate zero α reached up tothe 17th decimal place is reflected. These results of our five examples are obtained from computerhardware configuration with T2080 1.73 GHz CPU and 2.5 GB memory.

Example 1. There is a parametric curve c(t) = ( f1(t), f2(t), f3(t)) = (6t7 + t5, 5t8 + 3t6, 10t12 + 8t8 +

6t6 + 4t4 + 2t2 + 3), t ∈ [−2, 2] in three-dimensional Euclidean space and a test point p = (p1, p2, p3) =

(2.0, 4.0, 2.0). Using our method, the corresponding orthogonal projection parametric value is α = 0.0, the initialvalues t0 are 0,2,4,5,6,8,9,10, respectively. For each initial value, the iteration process runs 10 times and then10 different iteration times in nanoseconds, respectively. In Table 1, the average run time of our method for eightdifferent initial values are 536,142, 77,622, 101,481, 119,165, 126,502, 142,393, 150,801, 156,413 nanoseconds,respectively. Finally, the overall average running time is 176,315 nanoseconds (see Figure 6). If test pointp is (2.0, 2.0, 2.0), the corresponding orthogonal projection parametric value is α = 0.0, we replicate theprocedure using our method and report the results in Table 2. In Table 2, the average running time of ourmethod for 8 different initial values are 627,996, 89,992, 119,241, 139,036, 148,269, 167,364, 167,364, 178,554nanoseconds, respectively. Finally, the overall average running time is 205,228 nanoseconds (see Figure 7).However, for the above two cases, the H-H-H method does not converge for any initial iterative value.

Because of a singular point on the parametric curve, we have also added some pre-processing stepsbefore our method. (1) Find the singular point (0,0,3) and the corresponding parametric value 0 by usingthe methods [21,46–48]. (2) Using our method, the orthogonal projection points of test points (2,4,2) and(2,2,2) and their corresponding parameter values 0 and 0 are calculated, respectively. (3) From all candidatepoints(singular point and orthogonal projection point), corresponding point is selected so that the distancebetween the corresponding point and the test point is the minimum one. In Figure 6, the blue point denotessingular point (0,0,3), which is also the orthogonal projecting point of the test point (2,4,2). This is the same forthe blue point in Figure 7.

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Table 1. Running time for different initial values of Example 1 by our method with test point p = (2.0,4.0, 2.0).

t0 0 2 4 5 6 8 9 10

α 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 498,454 75,487 105,563 116,470 123,031 134,941 154,253 156,8722 555,709 81,629 108,064 117,762 125,946 140,940 153,468 155,8303 509,173 82,824 100,744 111,206 134,367 141,705 150,013 158,7154 564,222 77,465 96,721 114,757 129,128 173,027 150,320 158,5805 502,986 81,028 97,142 118,535 120,668 132,856 155,335 149,4376 553,198 79,520 104,307 120,795 129,351 150,085 151,073 143,0657 576,814 74,268 100,231 115,002 132,322 139,919 154,754 159,0148 524,848 81,982 99,604 115,263 122,401 139,345 143,568 175,1699 528,848 71,228 103,186 140,023 122,040 135,006 145,434 154,01610 547,161 70,789 99,247 121,834 125,766 136,103 149,790 153,435Average 536,142 77,622 101,481 119,165 126,502 142,393 150,801 156,413

Total Average 176,315


t0 0 2 4 5 6 8 9 10

α 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 595,515 92,371 119,904 135,660 148,751 162,758 171,535 177,3552 648,825 91,746 119,348 135,284 148,531 162,541 171,333 176,4313 595,772 91,633 119,248 135,322 148,222 162,240 171,095 176,5014 648,472 91,565 119,139 135,355 148,165 191,884 171,366 176,3955 595,856 91,556 119,168 135,406 148,144 162,224 171,417 176,5076 648,305 91,532 119,018 135,316 148,169 183,342 171,413 176,4737 647,406 91,587 119,069 135,283 148,197 162,291 171,282 176,3978 595,423 91,617 119,247 135,140 14,8101 162,116 171,342 196,5299 646,551 83,167 119,135 172,412 148,149 162,148 171,313 176,39010 657,838 83,147 119,131 135,179 148,259 162,094 171,609 176,557Average 627,996 89,992 119,241 139,036 148,269 167,364 171,371 178,554


Figure 6. Geometric illustration for the test point p = (2.0, 4.0, 2.0) of Example 1.

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Figure 7. Geometric illustration for the test point p = (2.0, 2.0, 2.0) of Example 1.

Example 2. There is a spatial quartic quasi-rational Bézier curve c(t) = ( f1(t), f2(t), f3(t)) =

(u(t)a(t)

,v(t)a(t)

,w(t)a(t)

), where u(t) = 2t4 + 3t3 + 3t2 + 12t + 1, v(t) = 4t4 + 3t3 + 7t2 + 7t + 21, w(t) =

5t4 + t3 + 9t2 + 11t + 13, a(t) = 4t4 + 8t3 + 17t2 + 15t + 6, t ∈ [−2, 2] and a test point p = (p1, p2, p3) =

(1.0, 3.0, 5.0). The corresponding orthogonal projection parametric value α are −1.4118250062741212,−0.61917136491841674, −0.059335038305820650, 1.8493434997820080, respectively. Using our method,the initial values t0 are −2.4,−2.1,−2.0,−1.8,−1.6,−1.2,−1.0,−0.8, respectively. For each initial value, theiteration process runs 10 times and then 10 different iteration times in nanoseconds, respectively. From Table 3,the average running time of our method for eight different initial values are 85,344, 93,936, 79,424, 62,643,54,482, 22,982, 25,654, 26,868 nanoseconds, respectively. Finally, the overall average running time is 56,417nanoseconds (see Figure 8). If test point p is (2.0, 4.0, 8.0), the corresponding orthogonal projection parametricvalue α are −1.2589948653798823, −0.62724968160147096,−0.14597283439336865, 1.8584532894110559,respectively. We firstly replicate the procedure using our method and report the results in Table 4. From Table 4,the average running time of our method for eight different initial iterative values are 101,436, 109,001, 95,061,77,563, 62,366, 27,054, 29,587, 32,501 nanoseconds, respectively. Finally, the overall average running timeis 66,821 nanoseconds (see Figure 9). We then replicate the procedure using the algorithm [26] and report theresults in Table 5. From Table 5, the average running time of the algorithm [26] for eight different initial valuesare 619,772, 654,281, 584,653, 467,856, 384,393, 163,225, 183,257, 195,013 nanoseconds, respectively. Finally,the overall average running time is 406,556 nanoseconds. However, for the above two cases, the H-H-H methoddoes not converge for any initial value.

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t0 −2.4 −2.1 −2 −1.8 −1.6 −1.2 −1 −0.8

α −1.4118 0.61917 −1.4118 0.61917 −0.059 −0.059 1.84934 1.849341 88,695 90,501 75,137 68,499 52,014 24,731 26,295 28,4442 89,958 91,254 79,411 64,563 54,321 22,014 26,278 28,0243 83,956 95,063 79,553 63,237 54,683 22,733 24,813 28,7604 83,623 96,033 82,022 68,075 51,098 23,270 24,573 26,7075 83,368 95,700 76,197 63,518 51,752 22,321 24,644 26,5866 83,631 97,303 80,984 62,608 53,473 21,658 24,009 28,2097 87,286 94,655 78,483 66,844 52,277 23,502 25,554 28,7258 87,150 96,316 79,215 64,333 51,554 23,217 26,234 28,2959 86,300 89,399 94,487 66,665 50,279 23,190 25,791 26,16010 89,761 96,377 82,362 64,371 50,367 22,332 23,929 27,273Average 85,344 93,936 79,424 62,643 54,482 22,982 25,654 26,868



t0 −2.4 −−2.1 −−2 −1.8 −1.6 −1.2 −1 −0.8

α −0.6272 −0.1459 −0.6272 −1.2589 −0.1459 1.858 −1.2589 1.8581 101,366 109,667 92,799 77,983 62,865 29,460 29,755 32,6492 102,027 108,844 92,709 77,477 62,269 27,177 29,555 32,4583 101,526 109,010 92,709 77,587 62,284 26,885 29,619 32,5384 101,266 108,909 92,724 77,441 62,374 26,785 29,557 32,4785 101,346 108,944 92,714 77,386 62,214 26,691 29,559 32,5056 101,315 108,990 92,764 77,557 62,334 26,731 29,564 32,4977 101,415 108,834 92,614 77,582 62,415 26,720 29,573 32,5128 101,306 108,945 92,528 77,461 62,309 26,715 29,548 32,4939 101,562 108,954 116,107 77,542 62,284 26,684 29,549 32,42910 101,235 108,910 92,939 77,616 62,314 26,690 29,595 32,451Average 101,436 109,001 95,061 77,563 62,366 27,054 29,587 32,501


Table 5. Running time for different initial values of Example 2 by the algorithm [26].

t0 −2.4 −2.1 −2.0 −1.8 −1.6 −1.2 −1.0 −0.8

α −0.6272 −0.1459 −0.6272 −1.2589 −0.1459 1.858 −1.2589 1.8581 633,173 660,734 566,675 470,236 391,687 171,352 175,965 198,5432 597,065 628,741 565,012 485,368 367,539 161,649 185,457 197,7983 652,494 675,268 600,951 463,899 396,359 163,879 188,682 187,1284 649,281 653,066 573,597 460,967 385,325 156,876 182,979 195,2145 622,109 687,282 568,766 472,217 402,669 170,876 189,508 202,5406 633,737 627,667 562,864 490,735 374,340 165,445 175,457 191,0377 584,705 637,608 563,523 468,230 395,411 163,631 175,676 187,5398 607,439 693,001 585,948 449,706 400,728 161,467 189,216 187,4339 637,036 639,359 671,613 444,834 359,918 157,235 188,119 195,86710 580,678 640,082 587,577 472,368 369,954 159,834 181,510 207,033Average 619,772 654,281 584,653 467,856 384,393 163,225 183,257 195,013


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Figure 8. Geometric illustration for the first case of Example 2.

Figure 9. Geometric illustration for the second case of Example 2.

Example 3. There is a parametric curve c(t) = ( f1(t), f2(t), f3(t), f4(t), f5(t)) =

(cos(t), sin(t), t, cos(t), sin(t)), t ∈ [−2, 2] in five-dimensional Euclidean space and a test pointp = (p1, p2, p3, p4, p5) = (3.0, 4.0, 5.0, 6.0, 7.0). Using our method, the corresponding orthogonalprojection parametric value is α = 1.1587403612284800, the initial values t0 are −10,−8,−6,−4, 4, 8, 12, 16,respectively. For each initial value, the iteration process runs 10 times and then 10 different iteration timesin nanoseconds, respectively. In Table 6, the average running time of our method for eight different initialvalues are 391,013, 424,444, 391,092, 249,376, 115,617, 170,212, 179,465, 196,912 nanoseconds, respectively.Finally, the overall average running time is 264,766 nanoseconds. If test point p is (30.0, 40.0, 50.0, 60.0, 70.0),the corresponding orthogonal projection parametric value α is 1.2352898417860202. We then replicate theprocedure using our method and report the results in Table 7. In Table 7, the average running time of our methodfor eight different initial values are 577, 707, 485, 417, 460, 913, 289, 232, 133, 661, 199, 470, 211, 915, 229, 398nanoseconds, respectively. Finally, the overall average running time is 323,464 nanoseconds. However, for theabove parametric curve and many test points, the H-H-H method does not converge for any initial value.

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Table 6. Running time for different initial values of Example 3 by our method with test point p = (3, 4,5, 6, 7).

t0 −10 −8 −6 −4 4 8 12 16

α 1.15874 1.15874 1.15874 1.15874 1.15874 1.15874 1.15874 1.158741 407,427 425,388 387,337 306,115 110,887 161,079 187,144 184,1192 417,729 446,171 398,801 341,895 121,148 169,115 169,954 194,6713 420,894 390,507 383,308 260,183 115,033 165,103 171,989 198,8844 383,836 421,365 427,391 242,641 109,521 161,121 179,152 195,7145 373,696 421,551 373,171 266,584 120,844 187,930 179,184 186,3096 374,791 445,114 373,974 242,889 119,449 183,082 180,269 201,4877 381,353 408,011 402,073 216,762 109,054 162,402 172,013 188,2068 398,662 442,008 373,328 194,821 119,236 192,990 180,472 197,2999 364,491 417,139 396,843 230,070 110,243 164,273 204,410 196,16310 387,246 427,188 394,694 191,799 120,759 155,029 170,059 226,270Average 391,013 424,444 391,092 249,376 115,617 170,212 179,465 196,912


Table 7. Running time for different initial values of Example 3 by our method with test point p = (30,40, 50, 60, 70).

t0 −10 −8 −6 −4 4 8 12 16

α 1.235289 1.235289 1.235289 1.235289 1.235289 1.235289 1.235289 1.2352891 1,190,730 475,499 453,879 369,551 133,651 191,093 208,202 223,6952 1,031,760 500,975 486,534 380,881 133,638 190,959 208,490 236,6373 482,018 475,395 450,480 297,272 133,674 199,528 208,292 223,3124 428,081 475,588 475,100 277,356 133,635 186,919 208,438 223,8025 455,282 475,033 448,776 296,510 133,535 220,570 208,139 223,4716 428,321 499,776 448,617 277,353 133,590 220,625 208,046 223,2137 428,246 474,978 474,667 247,245 133,620 192,326 208,101 230,7918 453,374 502,500 448,503 235,415 133,594 220,635 208,087 223,1839 426,949 474,816 474,167 275,526 133,546 198,204 245,226 223,21310 452,306 499,605 448,409 235,207 134,128 173,843 208,127 262,661Average 577,707 485,417 460,913 289,232 133,661 199,470 211,915 229,398


Example 4. (Reference to [6]) There is a parametric curve c(t) = ( f1(t), f2(t)) = (t2, sin(t)), t ∈[−3, 3] in two-dimensional Euclidean space and a test point p = (p1, p2) = (1.0, 2.0). The correspondingorthogonal projection parametric value is α = 1.1063055095030472. Using our method, the initial values t0 are−100,−4, 5, 7, 8, 10, 11, 100, respectively. For each initial value, the iteration process runs 10 times and then10 different iteration times in nanoseconds, respectively. In Table 8, the average running time of our methodfor eight different initial iterative values are 62,816, 35,042, 27,648, 43,122, 21,625, 38,654, 21,518, 72,917nanoseconds, respectively. Finally, the overall average running time is 40,418 nanoseconds (see Figure 10).Implementing the same procedure, the overall average running time given by the H-H-H method is 231,613nanoseconds in Table 9, while the overall average running time given by the second order method [6] is 847,853nanoseconds in Table 10. Thus, our method is faster than the H-H-H method [38–40] and the second ordermethod [6].

Figure 10. Geometric illustration for Example 4.

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Table 8. Running time for different initial values of Example 4 by our method.

t0 −100 −4 5 7 8 10 11 100

α 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.1063051 63,345 35,580 27,069 41,551 22,304 36,858 21,478 72,2572 63,192 36,203 28,160 41,733 20,042 38,680 20,338 71,6203 61,306 33,833 27,400 44,198 23,078 37,704 23,757 73,1084 66,627 34,502 26,014 44,160 21,147 39,374 22,530 70,1545 62,583 35,053 29,275 42,800 20,817 39,339 23,046 73,1896 63,957 34,398 25,650 42,282 22,184 37,376 20,070 75,8727 60,865 35,929 28,944 42,134 19,964 40,078 21,943 71,6088 63,522 35,427 27,578 41,688 23,650 39,456 21,076 76,2839 60,551 35,508 28,563 44,542 20,280 38,463 20,596 71,78110 62,216 33,987 27,830 46,130 22,781 39,209 20,349 73,296Average 62,816 35,042 27,648 43,122 21,625 38,654 21,518 72,917


Table 9. Running time for different initial values of Example 4 by the H-H-H method.

t0 −100 −4 5 7 8 10 11 100

α 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.1063051 424,579 357,276 443,858 179,583 176,984 175,859 175,249 178,4452 425,680 358,510 179,137 177,849 182,701 176,665 176,463 207,1643 359,794 356,912 180,000 180,472 177,867 179,743 178,929 179,3724 371,119 357,214 179,567 179,804 184,542 177,675 177,854 179,6515 358,128 358,119 232,337 179,285 179,113 175,632 177,690 181,9766 358,470 357,893 179,985 179,941 178,600 178,289 178,565 181,8687 358,083 359,391 178,815 177,857 177,613 178,014 177,385 179,3618 477,393 357,011 178,029 179,525 175,684 176,000 175,413 180,9669 356,254 359,356 176,148 178,581 176,351 177,024 185,103 180,01310 356,801 359,773 213,327 177,252 176,993 178,060 177,655 181,427Average 384,630 358,146 214,120 179,015 178,645 177,296 178,031 183,024


Table 10. Running time for different initial values of Example 4 by the Algorithm [6].

t0 −100 −4 5 7 8 10 11 100

α 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.106305 1.1063051 681,353 107,102 119,083 120,328 122,504 115,181 113,566 542,1162 725,571 124,514 136,810 121,111 116,824 111,116 117,466 5,250,4813 669,249 111,052 122,151 125,261 124,865 116,105 120,309 5,523,8054 713,982 112,146 131,494 118,104 121,099 111,410 118,658 5,407,1665 699,433 111,347 118,830 121,003 118,694 115,182 124,917 5,259,4126 693,396 113,323 116,046 109,176 108,194 111,420 117,342 5,508,0497 691,375 114,667 115,748 123,330 127,812 118,635 119,208 5,348,5178 663,125 107,484 127,493 120,134 116,818 111,717 117,079 5,446,7039 731,148 128,918 122,897 120,947 120,985 113,777 125,463 5,251,58010 676,286 128,567 130,775 118,031 116,725 111,095 108,275 5,356,125Average 694,492 115,912 124,133 119,743 119,452 113,564 118,228 5,377,300


Example 5. (Reference to [6]) There is a parametric curve c(t) = ( f1(t), f2(t)) = (t, sin(t)), t ∈ [−3, 3]in two-dimensional Euclidean space and a test point p = (p1, p2) = (1.0, 2.0), the corresponding orthogonalprojection parametric value is α = 1.2890239979093887. Using our method, the initial values t0 are−100,−4, 5, 7, 8, 10, 11, 100, respectively. For each initial value, the iteration process runs 10 times andthen 10 different iteration time in nanoseconds, respectively. In Table 11, the average running time of ourmethod for eight different initial values are 50, 579, 28, 238, 22, 687, 34, 974,17, 781, 31, 186, 17, 210, 59, 116nanoseconds, respectively. Finally, the overall average running time is 32,721 nanoseconds (see Figure 11).We then replicate the procedure using the second order method [6] and report the results in Table 12. In Table 12,the average running time of the second order method [6] for 8 different initial values are 320, 035,182, 451,147, 031, 235, 779, 112, 090, 200, 431, 113, 284, 369, 294 nanoseconds, respectively. Finally, the overall average

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running time is 210,049 nanoseconds. In addition, we compare the iterations by different methods where the NCdenotes non-convergence in Table 13.

Table 11. Running time for different initial values of Example 5 by our method.

t0 −100 −4 5 7 8 10 11 100

α 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.289021 52,010 27,426 21,791 33,323 18,399 29,551 16,486 58,9952 50,335 29,269 23,949 32,810 15,820 30,342 16,066 58,0803 49,047 26,841 23,061 37,063 19,611 31,569 19,756 57,4584 52,651 29,124 21,403 33,838 17,472 33,295 18,583 54,5665 49,871 29,814 25,062 35,870 16,655 32,949 18,304 61,8606 53,651 28,678 19,550 35,731 18,373 31,429 16,342 59,5707 47,275 28,115 24,177 35,456 16,933 30,510 18,010 59,0428 49,982 27,896 22,639 34,292 19,927 30,959 16,449 63,6529 49,704 29,359 22,502 34,164 17,274 30,391 16,044 61,37310 51,268 25,859 22,736 37,190 17,342 30,864 16,060 56,564Average 50,579 28,238 22,687 34,974 17,781 31,186 17,210 59,116


Table 12. Running time for different initial values of Example 5 by the Algorithm [6].

t0 −100 −4 5 7 8 10 11 100

α 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.289021 308,942 191,002 152,199 235,287 114,568 199,404 110,512 379,7712 348,554 175,800 146,728 232,260 102,698 190,860 101,754 352,8343 311,680 190,863 148,384 242,131 118,602 207,376 125,517 408,9784 332,421 166,849 145,131 234,536 102,795 198,956 113,523 370,8265 319,660 185,059 160,358 235,072 108,557 211,429 119,911 350,1886 329,882 177,252 132,242 233,702 120,945 199,978 107,366 363,2997 304,977 200,038 151,398 229,166 102,315 220,162 122,013 354,4668 326,645 171,624 137,588 228,181 113,627 195,782 108,512 369,8999 291,369 191,878 156,871 247,614 108,418 189,534 112,319 363,90510 326,221 174,148 139,415 239,836 128,377 190,831 111,411 378,781Average 320,035 182,451 147,031 235,779 112,090 200,431 113,284 369,294


Table 13. Comparison of iterations by different methods in Example 5.

t0 −100.0 −4.0 5.0 7.0 8.0 10.0 11.0 100.0

α 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902 1.28902H-H-H method [38–40] NC NC NC NC NC NC NC NCSecond order method [6] 75 30 32 32 33 29 31 101Newton’s method NC NC NC NC NC NC NC NCOur method 15 19 17 17 15 17 15 23

Figure 11. Geometric illustration for Example 5.

Remark 2. From the results of five examples, the overall average running time of our method is 145.5 μs.From the results of Table 9, the overall average running time of the H-H-H method is 231.6 μs. From resultsof six examples in [26], the overall average running time of the algorithm [1] is 680.8 μs. From results of

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six examples in [26], the overall average running time of the algorithm [14] is 1270.8 μs. From results ofTable 5, the overall average running time of the algorithm [26] is 406.6 μs. From results of Tables 10 and 12,the overall average running time of the algorithm [6] is 528.9 μs. Table 14 displays time comparison for thesealgorithms. In short, the robustness and efficiency of our method are more superior to those of the existingalgorithms [1,6,14,26,38–40].

Table 14. Time comparison of various algorithms.

Algorithms Ours H-H-H Algorithm [1] Algorithm [14] Algorithm [6] Algorithm [26]

Time (μs) 145.5 231.6 680.8 1270.8 528.9 406.6

Remark 3. For general parametric curve containing the elementary functions, such as sin(t),cos(t), et,ln t, arcsin t, arccos t, etc., it is very difficult to transform general parametric curve into Bézier-type curve.In contrast, our method can deal with the general parametric curve containing the elementary functions.Furthermore, the convergence of our method does not depend on the initial value. From Table 13, only theH-H-H method or the Newton’s method can not ensure convergence, while our method can ensure convergence.For multiple solutions of orthogonal projection, our approach works as follows:(1) The parameter interval [a, b] of parametric curve c(t) is divided into M identical subintervals.(2) An initial value is selected randomly in each interval.(3) Using our method and using each initial parametric value, do iterations, respectively. Suppose that theiterative parametric values are α1, α2, . . . , αM, respectively.(4) Calculate the local minimum distances d1, d2, . . . , dM, where di = ‖p− c(αi)‖.(5) Seek the global minimum distance d = ‖p− c(α)‖ from {‖p− c(a)‖ , d1, d2, . . . , dM, ‖p− c(b)‖}.

If we are to solve all solutions as far as possible, we urge the positive integer M to be as large as possible.

We use Example 2 to illustrate how the procedure works, where, for t ∈ [−2, 2], three parametervalues are −1.4118250062741212, −0.61917136491841674, 1.8493434997820080, respectively. It is easyto find that the projection point with the parameter value −0.61917136491841674 will be the one withminimum distance, whereas other projection points without these parameter values can not be the onewith minimum distance. Thus, only the orthogonal projection point with minimum distance remainsafter the procedure to select multiple orthogonal projection points.

Remark 4. We have done many test examples including five test examples. In the light of these test results,our method has good convergent properties for different initial values, namely, if initial value is t0, then thecorresponding orthogonal projection parametric value α for the orthogonal projection point of the test point p issuitable for one inequality relationship ∣∣⟨p− c(α), c′(α)

⟩∣∣ < 10−17. (21)

This indicates that the inequality relationship satisfies requirements of Equation (4). This shows thatconvergence of our method does not depend on the initial value. Furthermore, our method is robust and efficient,which is satisfied with the previous two of ten challenges proposed by [50].

5. Conclusions

This paper discusses the problem related to a point orthogonal projection onto a parametric curvein an n-dimensional Euclidean space on the basis of the H-H-H method, combining with a fixed pointtheorem of Newton’s method. Firstly, we run the H-H-H method. If the current iterative parametricvalue from the H-H-H method is satisfied with the convergence condition of the Newton’s method,we then go to the method to increase the convergence rate. Otherwise, we continue the H-H-H methodto generate the iterative parametric value with satisfaction of the local convergence condition bythe Newton’s method, and we then go to the previous step. Then, we run to the end of the whole

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process. The presented procedures ensure the convergence of our method and it does not depend onthe initial value. Analysis of convergence demonstrates that our method is second order convergent.Some numerical examples confirm that our method is more efficient and performs better than othermethods, such as the algorithms [1,6,14,26,38–40].

In this paper, our discussion focuses the algorithms in the parametric curve C2. For the parametriccurve being C0,C1, piecewise curve or having singular points, we only present a preliminary idea.However, we have not completely implemented an algorithm for this kind of spline with low continuity.In the future, we will try to construct several brand new algorithms to handle the kind of spline withlow continuity such that they can ensure very good robustness and efficiency. In addition, we also tryto extend this idea to handle point orthogonal projecting onto implicit curves and implicit surfacesthat include singularity points. Of course, the realization of these ideas is of great challenge. However,it is of great value and significance in practical engineering applications.

Author Contributions: The contribution of all the authors is the same. All of the authors team up to develop thecurrent draft. J.L. is responsible for investigating, providing methodology, writing, reviewing and editing thiswork. X.L. is responsible for formal analysis, visualization, writing, reviewing and editing of this work. F.P. isresponsible for software, algorithm and program implementation to this work. T.C. is responsible for validationof this work. L.W. is responsible for supervision of this work. L.H. is responsible for providing resources, writing,and the original draft of this work.

Funding: This research was funded by the National Natural Science Foundation of China Grant No. 61263034,the Feature Key Laboratory for Regular Institutions of Higher Education of Guizhou Province Grant No.2016003, the Training Center for Network Security and Big Data Application of Guizhou Minzu UniversityGrant No. 20161113006, the Key Laboratory of Advanced Manufacturing Technology, Ministry of Education,Guizhou University Grant No. 2018479, the National Bureau of Statistics Foundation Grant No. 2014LY011,the Key Laboratory of Pattern Recognition and Intelligent System of Construction Project of Guizhou ProvinceGrant No. 20094002, the Information Processing and Pattern Recognition for Graduate Education Innovation Baseof Guizhou Province, the Shandong Provincial Natural Science Foundation of China Grant No.ZR2016GM24,the Scientific and Technology Key Foundation of Taiyuan Institute of Technology Grant No. 2016LZ02, the Fundof National Social Science Grant No. 14XMZ001 and the Fund of the Chinese Ministry of Education GrantNo. 15JZD034.

Acknowledgments: We take the opportunity to thank the anonymous reviewers for their thoughtful andmeaningful comments.


References

1. Ma, Y.L.; Hewitt, W.T. Point inversion and projection for NURBS curve and surface: Control polygonapproach. Comput. Aided Geom. Des. 2003, 20, 79–99. [CrossRef]

2. Piegl, L.; Tiller, W. Parametrization for surface fitting in reverse engineering. Comput.-Aided Des. 2001, 33,593–603. [CrossRef]

3. Yang, H.P.; Wang, W.P.; Sun, J.G. Control point adjustment for B-spline curve approximation.Comput.-Aided Des. 2004, 36, 639–652. [CrossRef]

4. Johnson, D.E.; Cohen, E. A Framework for efficient minimum distance computations. In Proceedings of theIEEE Intemational Conference on Robotics & Automation, Leuven, Belgium, 20 May 1998.

5. Pegna, J.; Wolter, F.E. Surface curve design by orthogonal projection of space curves onto free-form surfaces.J. Mech. Des. ASME Trans. 1996, 118, 45–52. [CrossRef]

6. Hu, S.M.; Wallner, J. A second order algorithm for orthogonal projection onto curves and surfaces.Comput. Aided Geom. Des. 2005, 22, 251–260. [CrossRef]

7. Besl, P.J.; McKay, N.D. A method for registration of 3-D shapes. IEEE Trans. Pattern Anal. Mach. Intell. 1992,14, 239–256. [CrossRef]

8. Mortenson, M.E. Geometric Modeling; Wiley: New York, NY, USA, 1985.9. Limaien, A.; Trochu, F. Geometric algorithms for the intersection of curves and surfaces. Comput. Graph.

1995, 19, 391–403. [CrossRef]10. Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical recipes. In C: The Art of Scientific

Computing, 2nd ed.; Cambridge University Press: New York, NY, USA, 1992.

455


11. Elber, G.; Kim, M.S. Geometric Constraint solver using multivariate rational spline functions. In Proceedingsof the 6th ACM Symposiumon Solid Modeling and Applications, Ann Arbor, MI, USA, 4–8 June 2001;pp. 1–10.

12. Patrikalakis, N.; Maekawa, T. Shape Interrogation for Computer Aided Design and Manufacturing; Springer:Berlin, Germany, 2001.

13. Polak, E.; Royset, J.O. Algorithms with adaptive smoothing for finite minimax problems. J. Optim.Theory Appl. 2003, 119, 459–484. [CrossRef]

14. Selimovic, I. Improved algorithms for the projection of points on NURBS curves and surfaces. Comput. AidedGeom. Des. 2006, 439–445. [CrossRef]

15. Zhou, J.M.; Sherbrooke, E.C.; Patrikalakis, N. Computation of stationary points of distance functions.Eng. Comput. 1993, 9, 231–246. [CrossRef]

16. Cohen, E.; Lyche, T.; Riesebfeld, R. Discrete B-splines and subdivision techniques in computer-aidedgeometric design and computer graphics. Comput. Graph. Image Process. 1980, 14, 87–111. [CrossRef]

17. Piegl, L.; Tiller, W. The NURBS Book; Springer: New York, NY, USA, 1995.18. Park, C.-H.; Elber, G.; Kim, K.-J.; Kim, G.-Y.; Seong, J.-K. A hybrid parallel solver for systems of multivariate

polynomials using CPUs and GPUs. Comput.-Aided Des. 2011, 43, 1360–1369. [CrossRef]19. Barton, M. Solving polynomial systems using no-root elimination blending schemes. Comput.-Aided Des.

2011, 43, 1870–1878.20. van Sosin, B.; Elber, G. Solving piecewise polynomial constraint systems with decomposition and a

subdivision-based solver. Comput.-Aided Des. 2017, 90, 37–47. [CrossRef]21. Barton, M.; Elber, G.; Hanniel, I. Topologically guaranteed univariate solutions of underconstrained

polynomial systems via no-loop and single-component tests. Comput.-Aided Des. 2011, 43, 1035–1044.22. Johnson, D.E.; Cohen, E. Distance extrema for spline models using tangent cones. In Proceedings of the 2005

Conference on Graphics Interface, Victoria, Canada, 9–11 May 2005.23. Li, X.W.; Xin, Q.; Wu, Z.N.; Zhang, M.S.; Zhang, Q. A geometric strategy for computing intersections of two

spatial parametric curves. Vis. Comput. 2013, 29, 1151–1158. [CrossRef]24. Liu, X.-M.; Yang, L.; Yong, J.-H.; Gu, H.-J.; Sun, J.-G. A torus patch approximation approach for point

projection on surfaces. Comput. Aided Geom. Des. 2009, 26, 593–598. [CrossRef]25. Chen, X.-D.; Xu, G.; Yong, J.-H.; Wang, G.Z.; Paul, J.-C. Computing the minimum distance between a point

and a clamped B-spline surface. Graph. Models 2009, 71, 107–112. [CrossRef]26. Chen, X.-D.; Yong, J.-H.; Wang, G.Z.; Paul, J.-C.; Xu, G. Computing the minimum distance between a point

and a NURBS curve. Comput.-Aided Des. 2008, 40, 1051–1054. [CrossRef]27. Oh, Y.-T.; Kim, Y.-J.; Lee, J.; Kim, Y.-S. Gershon Elber, Efficient point-projection to freeform curves and

surfaces. Comput. Aided Geom. Des. 2012, 29, 242–254. [CrossRef]28. Sederberg, T.W.; Nishita, T. Curve intersection using Bézier clipping. Comput.-Aided Des. 1990, 22, 538–549.

[CrossRef]29. Barton, M.; Juttler, B. Computing roots of polynomials by quadratic clipping. Comput. Aided Geom. Des. 2007,

24, 125–141.30. Li, X.W.; Wu, Z.N.; Hou, L.K.; Wang, L.; Yue, C.G.; Xin, Q. A geometric orthogonal pojection strategy for

computing the minimum distance between a point and a spatial parametric curve. Algorithms 2016, 9, 15.[CrossRef]

31. Mørken, K.; Reimers, M. An unconditionally convergent method for computing zeros of splines andpolynomials. Math. Comput. 2007, 76, 845–865. [CrossRef]

32. Li, X.W.; Wang, L.; Wu, Z.N.; Hou, L.K.; Liang, J.; Li, Q.Y. Hybrid second-order iterative algorithm fororthogonal projection onto a parametric surface. Symmetry 2017, 9, 146. [CrossRef]

33. Li, X.W.; Wang, L.; Wu, Z.N.; Hou, L.K.; Liang, J.; Li, Q.Y. Convergence analysis on a second order algorithmfor orthogonal projection onto curves. Symmetry 2017, 9, 210.

34. Chen, X.-D.; Ma, W.Y.; Xu, G.; Paul, J.-C. Computing the Hausdorff distance between two B-spline curves.Comput.-Aided Des. 2010, 42, 1197–1206. [CrossRef]

35. Chen, X.-D.; Chen, L.Q.; Wang, Y.G.; Xu, G.; Yong, J.-H.; Paul, J.-C. Computing the minimum distancebetween two Bezier curves. J. Comput. Appl. Math. 2009, 229, 294–301. [CrossRef]

36. Kim, Y.J.; Oh, Y.T.; Yoon, S.H.; Kim, M.S.; Elber, G. Efficient Hausdorff distance computation for freeformgeometric models in close proximity. Comput.-Aided Des. 2013, 45, 270–276. [CrossRef]

456


37. Sundar, B.R.; Chunduru, A.; Tiwari, R.; Gupta, A.; Muthuganapathy, R. Footpoint distance as a measure ofdistance computation between curves and surfaces. Comput. Graph. 2014, 38, 300–309. [CrossRef]

38. Hoschek, J.; Lasser, D. Fundamentals of Computer Aided Geometric Design; A. K. Peters: Natick, MA, USA, 1993.39. Hu, S.M.; Sun, J.G.; Jin, T.G.; Wang, G.Z. Computing the parameter of points on NURBS curves and surfaces

via moving affine frame method. J. Softw. 2000, 11, 49–53.40. Hartmann, E. On the curvature of curves and surfaces defined by normal forms. Comput. Aided Geom. Des.

1999, 16, 355–376. [CrossRef]41. Li, X.W.; Mu, C.L.; Ma, J.W.; Wang, C. Sixteenth-order method for nonlinear Equations. Appl. Math. Comput.

2010, 215, 3754–3758. [CrossRef]42. Liang, J.; Li, X.W.; Wu, Z.N.; Zhang, M.S.; Wang, L.; Pan, F. Fifth-order iterative method for solving multiple

roots of the highest multiplicity of nonlinear equation. Algorithms 2015, 8, 656–668. [CrossRef]43. Melmant, A. Geometry and Convergence of Euler’s and Halley’s Methods. SIAM Rev. 1997, 39, 728–735.

[CrossRef]44. Traub, J.F. A Class of Globally Convergent Iteration Functions for the Solution of Polynomial Equations.

Math. Comput. 1966, 20, 113–138. [CrossRef]45. Smietanski, M.J. A perturbed version of an inexact generalized Newton method for solving nonsmooth

equations. Numer. Algorithms 2013, 63, 89–106. [CrossRef]46. Chen, F.; Wang, W.-P.; Liu, Y. Computing singular points of plane rational curves. J. Symb. Comput. 2008, 43,

92–117. [CrossRef]47. Jia, X.-H.; Goldman, R. Using Smith normal forms and μ-bases to compute all the singularities of rational

planar curves. Comput. Aided Geom. Des. 2012, 29, 296–314. [CrossRef]48. Shi, X.-R.; Jia, X.-H.; Goldman, R. Using a bihomogeneous resultant to find the singularities of rational space

curves. J. Symb. Comput. 2013, 53, 1–25. [CrossRef]49. Burden, R.L.; Faires, J.D. Numerical Analysis, 9th ed.; Brooks/Cole Cengage Learning: Boston, MA, USA, 2011.50. Piegl, L.A. Ten challenges in computer-aided design. Comput.-Aided Des. 2005, 37, 461–470. [CrossRef]


457

mathematics

Article

How to Obtain Global Convergence Domains viaNewton’s Method for Nonlinear Integral Equations

José Antonio Ezquerro * and Miguel Ángel Hernández-Verón

Department of Mathematics and Computation, University of La Rioja, Calle Madre de Dios, 53,26006 Logrono, Spain; [email protected]* Correspondence: [email protected]

Received: 3 May 2019; Accepted: 14 June 2019; Published: 17 June 2019

Abstract: We use the theoretical significance of Newton’s method to draw conclusions about theexistence and uniqueness of solution of a particular type of nonlinear integral equations of Fredholm.In addition, we obtain a domain of global convergence for Newton’s method.

Keywords: Fredholm integral equation; Newton’s method; global convergence

1. Introduction

Integral equations are very common in physics and engineering, since a lot of problems of thesedisciplines can be reduced to solve an integral equation. In general, we cannot solve integral equationsexactly and are forced to obtain approximate solutions. For this, different numerical methods canbe used. So, for example, iterative schemes based on the homotopy analysis method in [1], adaptedNewton-Kantorovich schemes in [2] and schemes based on a combination of the Newton-Kantorovichmethod and quadrature methods in [3]. Besides, techniques based on using iterative methods arealso interesting, since the theoretical significance of the methods allows drawing conclusions aboutthe existence and uniqueness of solution of the equations. The use of an iterative method allowsapproximating a solution and, by analysing the convergence, proving the existence of solution, locatinga solution and even separating such solution from other possible solutions by means of results ofuniqueness. The theory of fixed point plays an important role in the development of iterative methodsfor approximating, in general, a solution of an equation and, in particular, for approximating a solutionof an integral equation.

In this work, we pay attention to the study of nonlinear Fredholm integral equations withnonlinear Nemytskii operators of type

x(s) = �(s) + λ∫ b

aK(s, t)H(x)(t) dt, s ∈ [a, b], λ ∈ R, (1)

where �(s) ∈ C[a, b], kernel K(s, t) of integral equation is a known function in [a, b] × [a, b], H isa Nemytskii operator [4] given by H : Ω ⊆ C[a, b] → C[a, b], such that H(x)(t) = H(x(t)) andH : R→ R is a derivable real function, and x(s) ∈ C[a, b] is the unknown function to find.

It is common to use the Banach Fixed Point Theorem [5–7] to prove the existence of a unique fixedpoint of an operator and approximate it by the method of successive approximations. Moreover, globalconvergence for the method is obtained in the full space. For this, we use that the operator involved isa contraction.

Our main aim of this work is to do a study of integral Equation (1) from Newton’s method,

xn+1 = xn − [F′(xn)]−1F(xn), n ≥ 0, with x0 given,



that has quadratic convergence, superior to the convergence of the method of successiveapproximations, which is linear. This study is similar to that of the Fixed Point Theorem for themethod of successive approximations. In addition, we obtain a domain of global convergence,B(x, R) = {x ∈ C[a, b] : ‖x − x‖ < R}, with x ∈ C[a, b], for Newton’s method. Also, we obtaina result of uniqueness of solution that separate the approximate solution from other possible solutions.To carry out this study, we develop a technique based on the use of auxiliary points, which allowsobtaining domains of global convergence, locating solutions of (1) and domains of uniqueness ofthese solutions.

On the other hand, ifH(x) = x, integral Equation (1) is linear and well-known, it is a Fredholmintegral equation of the second kind, which is connected with the eigenvalue problem represented bythe homogeneous equation

x(s) = λ∫ b

aK(s, t)x(t) dt, s ∈ [a, b],

and has non-trivial solutions x(s) �≡ 0 for the characteristic values or eigenvalues λ (the latter term issometimes reserved to the reciprocals ν = 1/λ) of kernel K(s, t) and every non-trivial solution of (1) iscalled characteristic function or eigenfunction corresponding to characteristic value λ. If Equation (1)is nonlinear, our results allow doing a study of the equation based on the values of parameter λ, whichis another important aim of our work.

2. Global Convergence and Uniqueness of Solution

If we are interested in proving the convergence of an iteration, we can usually follow threeways to do it: local convergence, semilocal convergence and global convergence. First, from someconditions on the operator involved, if we require conditions to the solution x∗, we establish a localanalysis of convergence and obtain a ball of convergence of the iteration, which, from the initialapproximation x0 lying in the ball, shows the accessibility to x∗. Second, from some conditions on theoperator involved, if we require conditions to the initial iterate x0, we establish a semilocal analysis ofconvergence and obtain a domain of parameters, which corresponds to the conditions required to theinitial iterate, so that the convergence of iteration is guaranteed to x∗. Third, from some conditionson the operator involved, the convergence of iteration to x∗ in a domain, and independently of theinitial approximation x0, is established and global convergence is called. Observe that the three studiesrequire conditions on the operator involved and requirement of conditions to the solution, to the initialapproximation, or to none of these, is what determines the way of analysis.

The local analysis of the convergence has the disadvantage that it requires conditions on thesolution and this is unknown. The global analysis of convergence, as a consequence of the absence ofconditions on the initial approximations and the solution, is very specific for the operators involved.

In this paper, we focus our attention on the analysis of the global convergence of Newton’s methodand, as a consequence, we obtain domains of global convergence for nonlinear integral Equation (1)and also locate a solution. For this, we obtain a ball of convergence, by using an auxiliary point, thatcontains a solution and guarantees the convergence of Newton’s method from any point of the ball.

Solving Equation (1) is equivalent to solving the equation F (x) = 0, where F : Ω ⊆ C[a, b] −→C[a, b] and

[F (x)](s) = x(s)− �(s)− λ∫ b

aK(s, t)H(x)(t) dt, s ∈ [a, b], λ ∈ R, n ∈ N. (2)

Then,

[F′(x)y](s) = y(s)− λ∫ b

aK(s, t)[H′(x)y](t) dt = λ

∫ b

aK(s, t)H′(x(t))y(t) dt.

459


As a consequence,‖F′(x)−F′(y)‖ ≤ K‖x− y‖,

where K = |λ|SL, L is such that ‖H′(x)−H′(y)‖ ≤ L‖x− y‖, for all x, y ∈ Ω, and S =∥∥∥∫ b

a K(s, t) dt∥∥∥.

From the Banach lemma on invertible operators, it follows

‖Γ‖ = ‖[F′(x)]−1‖ ≤ 11− |λ|S‖H′(x)‖ = β, ‖ΓF (x)‖ ≤ ‖x− u‖+ |λ|S‖H(x)‖

1− |λ|S‖H′(x)‖ = η.

provided that|λ|S‖H′(x)‖ < 1. (3)

Next, we give some properties that are used later.

Lemma 1. For operator (2), we have:

(a) ΓF (x) = ΓF (x) + (x− x) +∫ 1

0 Γ (F′(x + t(x− x))−F′(x)) (x− x) dt, with x ∈ Ω.(b) F (xn) =

∫ 10 (F′(xn−1 + t(xn − xn−1))−F′(xn−1)) (xn − xn−1) dt, with xn−1, xn ∈ Ω.

As a consequence of item (b) of Lemma 1, it follows, for xn−1, xn ∈ Ω,

‖F (xn)‖ ≤ K2‖xn − xn−1‖2.

From the last result, and taking into account the parameters obtained previously, we analyze thefirst iteration of Newton’s method, what leads us to the convergence of the method.

If x0 ∈ B(x, R), then

‖Γ0‖ = ‖[F′(x0)]−1‖ ≤ β

1− KβR= α, ‖Γ0F′(x)‖ ≤ 1

1− KβR.

provided thatKβR < 1. (4)

Moreover, from item (a) of Lemma 1, it follows

‖x1 − x0‖ ≤ ‖Γ0F′(x)‖‖ΓF (x0)‖ < η + R + KβR2/21− KβR

= δ,

and, from item (b) of Lemma 1, we have

‖x1 − x‖ = ∥∥−Γ0(F (x0) +F′(x0)(x− x0)

)∥∥ ≤ ‖Γ0F′(x)‖‖ΓF (x)‖+ KβR2/21− KβR

≤ 2η + KβR2

2(1− KβR),

so that x1 ∈ B(x, R), provided that2η + KβR2

2(1− KβR)≤ R. (5)

Observe now that condition (5) holds if

Kβη ≤ 1/6 and R ∈ [R−, R+],

where R− =1−√1−6Kβη

3Kβ and R+ =1+√

1−6Kβη

3Kβ are the two real positive roots of quadratic equation

2η − 2R + 3KβR2 = 0.

460


After that, if we assume that

‖xn − xn−1‖ < γ2n−2‖xn−1 − xn−2‖, (6)

‖xn − x‖ < 2η + KβR2

2(1− KβR)≤ R, (7)

where γ = Kαδ/2, for all n ≥ 2, and provided that condition (5) holds, it follows in the same way that

‖xn+1 − xn‖ < γ2n−1‖xn − xn−1‖, ‖xn+1 − x‖ < 2η + KβR2

2(1− KβR)≤ R,

so that (6) and (7) are true for all positive integers n by mathematical induction.In addition, γ < 1 if

3(KβR)2 − 10(KβR) + 2(2− Kβη) > 0, (8)

which is satisfied provided that

Kβη ≤ 1/6 and R <5−√13 + 6Kβη

3Kβ.

As a consequence, condition (4) holds. More precisely, we can establish the following result.

Lemma 2. There always exists R > 0, such that inequalities (4), (5) and (8) hold, if

(a) Kβη ≤ 0.1547 . . . and R ∈[

R−,5−√13+6Kβη

3Kβ

),

(b) Kβη ∈ [0.1547 . . . , 1/6) and R ∈ [R−, R+],

where R− =1−√1−6Kβη

3Kβ and R+ =1+√

1−6Kβη

3Kβ .

Proof. First, we prove item (a) of Lemma 2. Observe that R− <5−√13+6Kβη

3Kβ , since Kβη ≤ 0.1547 . . ., so

that[

R−,5−√13+6Kβη

3Kβ

)�= ∅. Moreover, as Kβη ≤ 0.1547 . . ., we have 3(Kβη)2 + 6(Kβη)− 1 ≤ 0 and,

as a consequence, R+ >5−√13+6Kβη

3Kβ and R ∈[

R−,5−√13+6Kβη

3Kβ

)⊂ [R−, R+], so that (5) and (8) hold.

Second, if Kβη ∈ [0.1547 . . . , 1/6), then 3(Kβη)2 + 6(Kβη)− 1 ≥ 0 and R+ <5−√13+6Kβη

3Kβ , sothat R ∈ [R−, R+]. Then, (5) and (8) hold.

Third, in both cases, KβR < 1 follows immediately, since R <5−√13+6Kβη

3Kβ in items (a) and (b) ofLemma 2.

2.1. Convergence

Now, we can establish the following result.

Theorem 1. Suppose that Kβη ≤ 1/6 and consider R > 0 satisfying item (a) or item (b) of Lemma 2 and suchthat B(x, R) ⊂ Ω. If condition (3) holds, then Newtons’s method is well-defined and converges to a solution x∗

of F (x) = 0 in B(x, R) from every point x0 ∈ B(x, R).

Proof. From (6) and γ < 1, we have ‖xn+1 − xn‖ < ‖xn − xn−1‖, for all n ∈ N, so that sequence{‖xn+1 − xn‖} is strictly decreasing for all n ∈ N and, therefore, sequence {xn} is convergent. If x∗ =limn→∞ xn, then F (x∗) = 0, by the continuity of F and ‖F (xn)‖ → 0 when n → ∞.

From Theorem 1, the convergence of Newton’s method to a solution of equation F (x) = 0 isguaranteed. Moreover, the best ball of location of the solution is B(x, R−) and the biggest ball of

461


convergence is B(x, R+) or B(

x,5−√13+6Kβη

3Kβ

), depending on the value of Kβη is: Kβη ≤ 0.1547 . . .

for the former and Kβη ∈ [0.1547 . . . , 1/6) for the latter.

2.2. Uniqueness of Solution

For uniqueness of solution, we establish the following result, where uniqueness of solution isproved in B(x, R).

Theorem 2. Under conditions of Theorem 1, solution x∗ of F (x) = 0 is unique in B(x, R).

Proof. Assume that w∗ is another solution of F (x) = 0 in B(x, R) such that w∗ �= x∗. If operatorQ =

∫ 10 F′(w∗ + t(x∗ − w∗)) dt is invertible, we have x∗ = w∗, since Q(w∗ − x∗) = F (w∗)−F (x∗).

Then, as

‖I − ΓQ‖ ≤ ‖Γ‖∫ 1

0‖F′(x)−F′(w∗ + t(x∗ − w∗))‖dt

≤ βK∫ 1

0‖x− (w∗ + t(x∗ − w∗))‖dt (9)

= βKR

< 1,

it follows that Q is invertible by the Banach lemma on invertible operators and uniquenessfollows immediately.

Notice that, from Theorems 1 and 3, the best ball of location of a solution of (1) is B(x, R−) and

the best ball of uniqueness of solution and the biggest ball of convergence is B(

x,5−√13+6Kβη

3Kβ

)or

B(x, R+), depending on the value of Kβη lies.Once given the uniqueness of solution in the domain of existence of solution B(x, R), we enlarge

such domain from the following theorem.

Theorem 3. Under conditions of Theorem 1, we have that the solution x∗ is unique in the domain B(x, �) ∩Ω,where � = 2

Kβ − R.

Proof. Assume that w∗ is another solution of F (x) = 0 in B(x, �) ∩ Ω such that w∗ �= x∗. Then,from (9), it follows

‖I − ΓQ‖ < βK∫ 1

0((1− t)� + tR)dt = 1.

and Q is again invertible by the Banach lemma on invertible operators.

Note that � > 0, since βKR < 1, and uniqueness of solution is obtained in the ball of global

convergence given in Theorem 1, since � = 2Kβ − R ≥ 5−√13+6Kβη

3Kβ , R+.

3. Example

Now, we apply the last result to the following nonlinear integral equation:

x(s) = s3 +1825

∫ 1

0s3t3x(t)2dt, s ∈ [0, 1]. (10)

For Equation (10), we have λ = 18/25 and S =∥∥∥∫ 1

0 s3t3 dt∥∥∥ = 1/4 with the max-norm. As

H(x)(t) = x(t)2, then L = 2. If we choose x(s) = s3, then condition (3) holds, since |λ|S‖H′(x)‖ =

9/25 < 1. Moreover, β = 25/16 and η = 9/32 and K = |λ|SL = 9/25, so that Kβη = 0.1582 . . . ∈

462


[R−, R+], where R− = 0.4590 . . . and R+ = 0.7261 . . .. Therefore, from Theorem 1, the convergenceof Newton’s method to a solution of Equation (10) is guaranteed and the best ball of location of thesolution is B(s3, 0.4590 . . .) and the biggest ball of convergence is B(s3, 0.7261 . . .). Furthermore, fromTheorem 3, it follows that the domain of uniqueness of solution is B(s3, 3.0965 . . .).

Next, we approximate a solution of Equation (10) by Newton’s method. After four iterations withstopping criterion ‖xn − xn−1‖∞ < 10−18, n ∈ N, we obtain solution shown in Table 1, where errors‖x∗ − xn‖ and sequence ‖F (xn)‖ are also shown. Observe from the last sequence that solution shownin Table 1 is a good approximation of the solution of Equation (10). Finally, we observe in Figure 1 thatsolution shown in Table 1 lies within the domain of location of solution found above.

Table 1. Approximated solution x∗(s) of (10), absolute errors and {‖F (xn)‖}.

n xn(s) ‖x∗ − xn‖ ‖F (xn)‖0 s3 8.4715 . . .× 10−2 7.2× 10−2

1 (1.0841121495327102 . . .)s3 6.0365 . . .× 10−4 5.0938 . . .× 10−4

2 (1.0847157717628998 . . .)s3 3.1090 . . .× 10−8 2.6233 . . .× 10−8

3 (1.0847158028530592 . . .)s3 8.2478 . . .× 10−17 6.9595 . . .× 10−17

4 (1.0847158028530593 . . .)s3

Figure 1. Approximated solution x∗(s) of (10) and domain of location of solution.

4. Study of the Integral Equation from Parameter λ

Next, we study the integral Equation (1) from the values of parameter λ.First, we observe that Kβη ≤ 1/6 if

6|λ|SL (‖x− �‖+ |λ|S‖H(x)‖) ≤ (1− |λ|S‖H′(x)‖)2 (11)

and condition (3) holds.Now, we analyze condition (11). Observe that (11) is satisfied if

• ‖H′(x)‖2 < 6L‖H(x)‖ and |λ| ∈ [0, μ+], where

μ+ =−(3L‖x− �‖+ ‖H′(x)‖) +√ΔS(6L‖H(x)‖ − ‖H′(x)‖2)

and Δ = 3L(3L‖x− �‖2 + 2‖x− �‖‖H′(x)‖+ 2‖H(x)‖).• ‖H′(x)‖2 > 6L‖H(x)‖ and |λ| ∈ [0, μ+] ∪ [μ−,+∞), where

μ− =−(3L‖x− �‖+ ‖H′(x)‖)−√ΔS(6L‖H(x)‖ − ‖H′(x)‖2)

.

• ‖H′(x)‖2 = 6L‖H(x)‖ and |λ| ≤ 12S(3L‖x− �‖+ ‖H′(x)‖) .

463


Second, once x is fixed, we have two chances: Kβη ≤ 0.1547 . . . or Kβη ∈ [0.1547 . . . , 1/6). If first

holds, then R ∈[

R−,5−√13+6Kβη

3Kβ

)and, if second does, then R ∈ [R−, R+].

Finally, as condition (3) is satisfied, then Newtons’s method is well-defined and converges to asolution x∗ of F (x) = 0 in B(x, R) from every point x0 ∈ B(x, R) by Theorem 1.

5. Application

Now, we apply the last study to the following particular Davis-type integral Equation [8]:

x(s) = s + λ∫ 1

0G(s, t)x(t)2dt, λ ∈ R, s ∈ [0, 1], (12)

where the kernel of (12) is a Green’s function defined as follows:

G(s, t) =

{(1− s)t, t ≤ s,

s(1− t), s ≤ t.

One can show that the function x(s) that satisfied Equation (12) is any solution of thedifferential equation

x′′(s) + λx(s)2 = 0,

that also satisfies the two-point boundary condition: x(0) = 0, x(1) = 1.For Equation (12), we have S =

∥∥∥∫ 10 G(s, t) dt

∥∥∥ = 1/8 with the max-norm and H(x)(t) = x(t)2.Therefore, L = 2 and condition (3) is reduced to |λ| < 4/‖x‖. In addition,

‖H′(x)‖2 = 4‖x‖2 < 12‖x‖2 = 6L‖H(x)‖

and, as a consequence,

|λ| ≤ μ+ =−(6‖x− s‖+ 2‖x‖) +√

12 (3‖x− s‖2 + 2‖x− s‖‖x‖+ ‖x‖2)

‖x‖2 .

After that, we choose x(s) = s and hence μ+ = 2(−1 +√

3) = 1.4641 . . ., so that |λ| ≤ 1.4641 . . .,that satisfies condition (3). In this case, from Theorem 1, we can guarantee the convergence of Newton’smethod to a solution of Equation (12) with λ such that |λ| ≤ 1.4641 . . . Moreover, once λ is fixed,depending on the value of Kβη, we can obtain the best ball of location of solution and the biggest ballof convergence.

Observe that we cannot apply Newton’s method directly, since we do not know the inverseoperator that is involved in the algorithm of Newton’s method. Then, we use a process of discretizationto transform (12) into a finite dimensional problem. For this, we use a Gauss–Legendre quadratureformula to approximate the integral of (12),

∫ 1

0φ(t) dt �

m

∑j=1

wjφ(tj),

where the m nodes tj and weights wj are known.Next, we denote the approximations x(ti) by xi, with i = 1, 2, . . . , m, so that (12) is equivalent to

the nonlinear system given by

xj = tj + λm

∑k=1

ajk x2k , k = 1, 2, . . . , m, (13)

464


where

ajk =

{wk (1− tj)tk, k ≤ j,

wk (1− tk)tj, k > j.

After that, we write system (13) compactly in matrix form as

F(x) ≡ x− v− λA y = 0, F : Rm −→ Rm, (14)

where

x = (x1, x2, . . . , xm)t , v = (t1, t2, . . . , tm)

t , A =(

ajk

)m

j,k=1, y =

(x2

1, x22, . . . , x2

m

)t.

Choose m = 8, λ = 7/5, x = v and hence K = 0.2471 . . ., β = 1.2179 . . ., η = 0.0665 . . . andKβη = 0.0200 . . . As Kβη < 0.1547 . . ., it follows, from Theorem 1, that the best ball of location ofsolution is B(v, 0.0686 . . .) and the biggest ball of convergence is B(v, 1.5259 . . .).

If the starting point for Newton’s method is x0 = v, the method converges to the solutionx∗ = (x∗1, x∗2, . . . , x∗8)t of system (14), which is shown in Table 2, after four iterations with stoppingcriterion ‖xn − xn−1‖∞ < 10−18, n ∈ N.

Table 2. Numerical solution x∗ of system (14) with λ = 7/5.

i x∗i i x∗i1 0.02267000. . . 5 0.65888692. . .2 0.11607746. . . 6 0.82291926. . .3 0.27057507. . . 7 0.93276524. . .4 0.46275932. . . 8 0.98797444. . .

Moreover, errors ‖x∗ − xn‖ and sequence {‖F(xn)‖} are shown in Table 3. Observe then thatvector shown in Table 2 is a good approximation of a solution of (14).

Table 3. Absolute errors and {‖F(xn)‖}.

n ‖x∗ − xn‖ ‖F(xn)‖0 6.7169 . . .× 10−2 9.6624 . . .× 10−1

1 6.4734 . . .× 10−4 5.3956 . . .× 10−4

2 6.0570 . . .× 10−8 5.0516 . . .× 10−8

3 5.4063 . . .× 10−16 4.5077 . . .× 10−16

Furthermore, as a solution of (12) satisfies x(0) = 0 and x(1) = 1, if values of Table 2 areinterpolated, an approximated solution is obtained, which is painted in Figure 2. Notice that thisapproximated solution lies in the domain of location of solution B(v, 0.0686 . . .) which is obtainedfrom Theorem 1.

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Figure 2. Solution x∗ of system (14) and domain of location of solution.

6. Conclusions

Following the idea of the Fixed Point Theorem for the method of successive approximations,we do an analysis for Newton’s method, use the theoretical significance of the method to prove theexistence and uniqueness of solution of a particular type of nonlinear integral equations of Fredholmand, in addition, obtain a domain of global convergence for the method that allows locating a solutionand separating it from other possible solutions. For this, we use a technique based on using auxiliarypoints. Moreover, we present a study of the nonlinear equations which is based on the real parameterinvolved in the equation.

Author Contributions: The contributions of the two authors have been similar. Both authors have workedtogether to develop the present manuscript.

Funding: This research was partially supported by Ministerio de Ciencia, Innovación y Universidades undergrant PGC2018-095896-B-C21.


References

1. Awawdeh, F.; Adawi, A.; Al-Shara, S. A numerical method for solving nonlinear integral equations.Int. Math. Forum 2009, 4, 805–817.

2. Nadir, M.; Khirani, A. Adapted Newton-Kantorovich method for nonlinear integral equations. J. Math. Stat.2016, 12, 176–181. [CrossRef]

3. Saberi-Nadja, J.; Heidari, M. Solving nonlinear integral equations in the Urysohn form byNewton-Kantorovich-quadrature method. Comput. Math. Appl. 2010, 60, 2018–2065.

4. Matkowski, J. Functional Equations and Nemytskii Operators. Funkc. Ekvacioj 1982, 25, 127–132.5. Berinde, V. Iterative Approximation of Fixed Point; Springer: New York, NY, USA, 2005.6. Ragusa, M.A. Local Hölder regularity for solutions of elliptic systems. Duke Math. J. 2002, 13, 385–397.

[CrossRef]7. Wang, H.; Zhang, L.; Wang, X. Fixed point theorems for a class of nonlinear sum-type operators and

application in a fractional differential equation. Bound. Value Probl. 2018, 2018, 140. [CrossRef]8. Davis, H.T. Introduction to Nonlinear Differential and Integral Equations; Dover: New York, NY, USA, 1962.


466

mathematics

Article

Numerical Solution of Heston-Hull-WhiteThree-Dimensional PDE with a High OrderFD Scheme

Malik Zaka Ullah

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected]

Received: 27 June 2019; Accepted: 1 August 2019; Published: 6 August 2019��

Abstract: A new numerical method for tackling the three-dimensional Heston–Hull–White partialdifferential equation (PDE) is proposed. This PDE has an application in pricing options when notonly the asset price and the volatility but also the risk-free rate of interest are coming from stochasticnature. To solve this time-dependent three-dimensional PDE as efficiently as possible, high orderadaptive finite difference (FD) methods are applied for the application of method of lines. It is derivedthat the new estimates have fourth order of convergence on non-uniform grids. In addition, it isproved that the overall procedure is conditionally time-stable. The results are upheld via severalnumerical tests.

Keywords: heston model; Hull–White; option pricing; PDE; finite difference (FD)

MSC: 41A25; 65M22

1. Introduction

To model different types of derivatives in finance, a common approach is to investigatethe connections of these factors to each other, formulated as a stochastic differential equation (SDEs).The factors could be the underlying asset, the volatility, domestic and foreign interest rates, etc., [1,2].As such, the important action of pricing option under different payoffs can be modeled and simulatedvia the SDEs or their corresponding partial differential equation (PDE) formulation.

However, a frequently occurring issue is that whatever the model becomes complicated and morerealistic, the procedure of having and representing its exact solution becomes harder, see, e.g., [3–5].

To discuss more and from the beginning, the classical model of Black–Scholes in pricing contractsdoes not cover and illustrate all the aspects of an option in a complete market, such as market risks,stochastic volatility (SV), and asymmetries seen in data of market, [6]. Some remedies to this well-knownmodel are via non-lognormal hypothesis for a SDE, that indicates some modifications of the volatilityand the underlying asset. We recall that Heston in [7] extended and improved the behavior of theBlack–Scholes model by involving more risky factor into the model, i.e., by considering the volatility tobe stochastic as well. Further discussions can be found at [6,8].

On the other hand, as long as the foreign exchange (FX) products are involved and a traderencounters a situation in which the interest rate is not anymore constant during the lifetimeof an option, then investigating and proposing an improved model, having stochastic rates of interest,such as the power-reverse dual-currency and the Heston–Cox–Ingersoll–Ross (HCIR) problems(refer to [9] and the references therein for more background).



1.1. Problem Formulation

The option pricing problem under the 3D Heston–Hull–White (HHW) model as a PDE modelis defined by [10]:

∂u(s, v, r, t)∂t

=12

s2v∂2u(s, v, r, t)

∂s2 +12

σ21 v

∂2u(s, v, r, t)∂v2 +

12

σ22

∂2u(s, v, r, t)∂r2

+ ρ12σ1sv∂2u(s, v, r, t)

∂s∂v+ ρ13σ2s

√v

∂2u(s, v, r, t)∂s∂r

+ ρ23σ1σ2√

v∂2u(s, v, r, t)

∂v∂r

+ rs∂u(s, v, r, t)

∂s+ κ(η − v)

∂u(s, v, r, t)∂v

+ a(b(T − t)− r)∂u(s, v, r, t)

∂r− ru(s, v, r, t).

(1)

Here κ > 0 shows the volatility adjustment speed to the analytical mean η > 0, while σ1, σ2, a aresome parameters. In addition, the correlation parameters are ρ12, ρ13, ρ23 ∈ [−1, 1], b is a time function.

In pricing under call options, the (terminal/)initial condition is given by [11,12]:

u(s, v, r, 0) = (s− E)+ , (2)

where the strike price is E. In a similar way, for a put option, it is given as follows:

u(s, v, r, 0) = (E− s)+ . (3)

As discussed in [13,14], the fair pricing procedure should be carried out by computationalschemes since the corresponding high-dimensional PDEs, constructed for such options, do not admitany analytical or semi-analytical solutions, see [15,16] for further background.

1.2. Novelties and Motivation

The contribution of this article reads in proposing a solution method via an un-equally spaced gridhaving a focus on the hot area in option pricing under the HHW PDE problem. Studying and codingmulti dimensional problems with discretization methods while the grid of points are non-uniformis a challenging and intensive task, but could clearly increase the accuracy of the approximate solutionby applying fewer numbers of grid nodes in contrasts to the uniform discretization. This reducesthe size of the discretized problem and is useful in practice.

To this aim, (1) is tackled by employing high order fourth-order finite difference approximations.We apply fourth order discretizations on a stencil having five and six non-equidistant nodes.Derivation and construction of fourth-order compact FD method for HHW PDE is new and usefulin practice.

In fact, the method-of-lines technique is considered to build a set of ODEs with time-varyingsystem matrix. All the side conditions are imposed therein as well. Thence, a method to march alongtime for the set of ODEs is provided in Section 3 and it is analytically illustrated that the presentednumerical procedure is conditionally time-stable when b is not changing by time.

Recalling that here adaptive FD formulas are constructed to hit some features simultaneously,viz., to be effective, results in sparse operators and being able to handling non-uniform grids.

Motivated by recent works in this field (see e.g., [17]), we aim at proposing higher order schemesfor the HHW equation on non-uniform meshes so as to increase the accuracy of obtained option priceswithout increasing the computational load so much. The novelties and contributions of our work aregiven below:

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• We propose fourth-order adaptive discretizations for the spatial variables.• The beauty of our scheme is the use of non-uniform grid of nodes with an adaptation

on the hotzone.• We provide a new stability bound for the resulting fully discretized set of equations when

pricing under HHW PDE using high order discretization methods along the spatial as well asthe temporal variables.

1.3. Grid Generation

The option pricing problem (1) is considered in the unbounded area

(s, v, r, t) ∈ Ω× (0, T], (4)

wherein Ω = [0,+∞) ×[0,+∞)× [0,+∞). For tackling the financial model numerically, one can takeinto account the following domain [18]:

Ω = [0, smax]× [0, vmax]× [−rmax, rmax], (5)

wherein smax, vmax, rmax are three positive real constants and assumed to be large enough.Since the PDE model is coercive (sometimes called degenerate) at v = 0, its payoff is non-smooth

at s = E, and the working domain has large width, thus it is requisite to use non-uniform meshes,at which the location of the nodes are not equally-spaced. This helps in producing results of higheraccuracy with adapting to the hotzone of the problem.

Let {si}mi=1 be a set of non-uniform nodes along s as follows [13,19]:

si = ϕ(ξi), 1 ≤ i ≤ m, (6)

where m > 1 and ξmin = ξ1 < ξ2 < · · · < ξm = ξmax are m equi–distant points with the followingcharacteristics: ξmin = sinh−1

(smin−sleft

d1

), ξint =

sright−sleftd1

, ξmax = ξint + sinh−1( smax−sright

d1

),

wherein smin = 0. Here d1 > 0 controls the density of the nodes around s = E. We also have:

ϕ(ξ) =

⎧⎪⎨⎪⎩sleft + d1 sinh(ξ), ξmin ≤ ξ < 0,sleft + d1ξ, 0 ≤ ξ ≤ ξint,sright + d1 sinh(ξ − ξint), ξint < ξ ≤ ξmax.

(7)

Throughout this work, we used the same value for d1 = E20 while sleft = max{0.5, exp{−0.25T}} × E,

[sleft, sright] ⊂ [0, smax], sright = E and smax = 14E.The nodes along v, i.e., {vj}n

j=1 are defined by:

vj = d2 sinh(ς j), 1 ≤ j ≤ n, (8)

where d2 > 0 gives the concentration around v = 0. In this work, we used d2 = vmax500 , where vmax = 10.

In addition, ς j are equally spaced points given by:

ς j = (j− 1)Δς, Δς =1

n− 1sinh−1

(vmax

d2

), (9)

for any 1 ≤ j ≤ n. The non-uniform nodes along r are defined as follows:

rk = d3 sinh(ζk), 1 ≤ k ≤ o, (10)

whereas d3 = rmax500 is a positive parameter and rmax = 1. We also have ζk = (k− 1)Δζ,

Δζ = 1o−1 sinh−1

(rmax

d3

). Note that denser mesh points in the important area could circumvent

469


the problems happening in solving (1), like non-smoothness of payoffs (2) and (3) at s = E,and the degeneracy at v = r = 0.

We state that a detailed study into possibly better choices for the involved parameters in meshgenerating may be interesting, but this is beyond the scope of the current research. Furthermore,the non-smoothness arising in the payoff would ruin the convergence rate of most derivativeapproximation particularly on uniform meshes and due to this, the application of non-uniformnodes is indispensable for efficient numerical solution of (1).

1.4. Manuscript Organization

The remaining parts of this work are organized as follows. In Section 2, the weights of the FDscheme over non-uniform grids (here we also call adaptive grids with special emphasis on the hot zone)are derived to attain the higher rate of convergence four.

Section 3 is devoted to the application of a sixth order Runge–Kutta time stepping methodto advance along time when semi-discretize the HHW PDE. We prove that the new procedureis time-stable conditionally based on the largest eigenvalue of the system matrix. Section 4 shows thatnumerical performances are more useful than the earlier schemes with quicker convergence behavior.Finally, some conclusions are drawn in Section 5.

2. Calculating the Weights of the High Order FD Scheme

In this section, by applying a methodology as in ([20], Chapters 3–4) or [21], but with more Taylorexpansion terms, we can construct fourth-order FD approximations on (non-uniform) grids.

Five points are required in estimating the first derivative as well as six points in approximatingthe second derivative in order to obtain a consistent fourth-order scheme throughout the discretizedmesh of points.

Without losing the generality, let us construct the weights in the one dimensional case.Then, the concept of tensors using Kronecker product may be applied easily to transfer the weightsto the appropriate dimensions. To this objective, consider a sufficiently smooth function g(s)and a grid as follows:

{s1, s2, · · · , sm−1, sm}. (11)

Consider the following five adjacent nodes:

{{si−2, g(si−2)}, {si−1, g(si−1)}, {si, g(si)}, {si+1, g(si+1)}, {si+2, g(si+2)}}, (12)

and calculate the interpolation polynomial p(z) going via the nodes and then its first derivative p′(z).At this moment, by employing a computer algebra system to do some symbolic computations

and setting z = si, we attain the FD estimate for the first derivative as follows:

g′(si) = αi−2g (si−2) + αi−1g (si−1) + αig (si) + αi+1g (si+1) + αi+2g (si+2) +O(

h4)

, (13)

470


where the maximum local grid spacing is h and we have

αi−2 = − Γi−1,iΓi,i+1Γi,i+2

Γi−2,i−1Γi−2,iΓi−2,i+1Γi−2,i+2,

αi−1 =Γi−2,iΓi,i+1Γi,i+2

Γi−2,i−1Γi−1,iΓi−1,i+1Γi−1,i+2,

αi =1

Γi−2,iΓi,i−1Γi,i+1Γi,i+2Ξ1,

αi+1 =Γi−2,iΓi,i−1Γi,i+2

Γi−2,i+1Γi+1,i−1Γi+1,iΓi+1,i+2,

αi+2 =Γi−2,iΓi,i−1Γi,i+1

Γi−2,i+2Γi+2,i−1Γi+2,iΓi+2,i+1,

(14)

using Γl,q = sl − sq and

Ξ1 =si−2(si−1(Γi+1,i + Γi+2,i)

+ 3s2i − 2(si+1 + si+2)si + si+1si+2) + si(−4s2

i + 3(si+1 + si+2)si

− 2si+1si+2) + si−1(3s2i − 2(si+1 + si+2)si + si+1si+2).

(15)

Recalling that the above procedure should be similarly done for the nodes {s1, s2, sm−1, sm},viz, to find the weighting coefficients with fourth order of convergence for such nodes, we shouldconsider the five adjacent points and then calculate the interpolating polynomial at that specific point.In this way, the sided FD formulas are constructed and used.

Similarly, FD estimates for the second derivative terms can be obtained applying a similarmethodology as above. To this objective, we consider a set of points as follows:

{{si−3, g(si−3)}, {si−2, g(si−2)}, {si−1, g(si−1)},

{si, g(si)}, {si+1, g(si+1)}, {si+2, g(si+2)}},(16)

and compute the second-derivative interpolating polynomial p′′(z) based on z. Now by taking intoaccount z = si in Mathematica [22], one obtains that:

g′′(si) =βi−3g (si−3) + βi−2g (si−2) + βi−1g (si−1)

+ βig (si) + βi+1g (si+1) + βi+2g (si+2) +O(

h4)

,(17)

where

βi−3 =Ξ2

Γi−3,i−2Γi−3,i−1Γi−3,iΓi−3,i+1Γi−3,i+2,

βi−2 =Ξ3


βi−1 =Ξ4


(18)

βi =Ξ5

Γi−3,iΓi,i−2Γi,i−1Γi,i+1Γi,i+2,

βi+1 =Ξ6

Γi−3,i+1Γi+1,i−2Γi+1,i−1Γi+1,iΓi+1,i+2,

βi+2 =Ξ7

Γi−3,i+2Γi+2,i−2Γi+2,i−1Γi+2,iΓi+2,i+1.

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Here, we have

Ξ2 =si−1(−6s2i + 4(si+1 + si+2)si − 2si+1si+2) + 2si(4s2

i

− 3(si+1 + si+2)si + 2si+1si+2) + si−2(−6s2i

+ 4(si+1 + si+2)si − 2si+1si+2 + si−1(4si − 2(si+1 + si+2))),

Ξ3 =2(si−3(si−1(Γi+1,i + Γi+2,i) + 3s2i − 2(si+1

+ si+2)si + si+1si+2) + si(−4s2i + 3(si+1 + si+2)si − 2si+1si+2)

+ si−1(3s2i − 2(si+1 + si+2)si + si+1si+2)),

Ξ4 =2(si−3(si−2(Γi+1,i + Γi+2,i) + 3s2i − 2(si+1

+ si+2)si + si+1si+2) + si(−4s2i + 3(si+1 + si+2)si − 2si+1si+2)

+ si−2(3s2i − 2(si+1 + si+2)si + si+1si+2)),

(19)

Ξ5 =2(si−2(si−1(Γi+1,i + Γi+2,i − si) + 6s2i − 3(si+1 + si+2)si + si+1si+2) + si−3(si−1(Γi+1,i

+ Γi+2,i − si) + si−2(Γi+1,i + Γi+2,i + si−1 − si) + 6s2i − 3si+1si − 3si+2si + si+1si+2)

+ si(2si(3si+1 − 5si) + si−1(6si − 3si+1)) + (3si(2si − si+1) + si−1(si+1 − 3si))si+2),

Ξ6 =2(si(si−1(2Γi,i+2 + si) + si(3si+2 − 4si)) + si−2(si(2Γi,i+2 + si)

+ si−1(Γi+2,i − si)) + si−3(2siΓi,i+2

+ si−1(Γi+2,i − si) + si−2(Γi−1,i + Γi+2,i) + s2i )),

Ξ7 =2(si(si−1(2Γi,i+1 + si) + si(3si+1 − 4si)) + si−2(si(2Γi,i+1

+ si) + si−1(Γi+1,i − si)) + si−3(2siΓi,i+1

+ si−1(Γi+1,i − si) + si−2(Γi−1,i + Γi+1,i) + s2i )).

Summarizing the following theorem has been established.

Theorem 1. As long as the function g is sufficiently smooth, the first and second derivative of the thisfunction can be approximated by five and six adjacent points respectively on non-uniform meshes,via the formulas (13) and (17).

Proof. The proof can be investigated by Taylor expansions as in the derivation in this section.It is hence omitted.

The procedure for obtaining the weights for the points {s1, s2, s3, sm−1, sm} to keep the fourthconvergence order should be investigated by the six adjacent points as described above but for thatspecific node.

It is noted that the formulations derived in (13) and (17) can be used for both uniformand nonuniform distribution of the discretization nodes, and can be simplified to more simplerformulations if the nodes are equidistant.

3. Application to Option Pricing under 3D HHW PDE

Considering the non-uniform nodes discussed in Section 1 along with the high order FDformulations calculated in Section 2, one is able to derive the differentiation matrices correspondingto the first and second derivatives of the function. These derivative matrices contains the weightsof the fourth order approximations and are sparse in general since they are banded matrices whosezero elements are much more than their non-zero elements. These feature would help us in solvingthe financial model (1) as would be observed later.

For multi-dimensional derivatives, a matrix is constructed such that this is done on the flatteneddata, and subsequently the Kronecker product of the matrices for the derivatives (in one-dimension)are being considered.

472


One way for imposing the impact of (13)–(17) is with matrices including the weights of (13)–(17),i.e., the non-equidistant second-order FD weights, as their elements. A matrix which shows an estimationto the differential operator is called as a matrix of differentiation [20]. Forming and implementingthe proposed scheme based on these matrices are invaluable aids for analysis.

Taking all the weights into consideration, the PDE (1) can be semi discretized to obtain:

∂U(t)∂t

= A(t)U(t), 0 ≤ t ≤ T, (20)

at which U(t) = (u1,1,1(t), u1,1,2(t), . . . , um,n,o−1(t), um,n,o(t)︸︷︷︸N elements

)∗, is the unknowns vector

and N = m× n× o. Noting that AN×N(t) is the coefficient of the problem (1) at which the boundarieshave not yet been imposed inside.

Here the boundaries along s are defined as follows [13]:

u(s, v, r, t) = 0, s = 0, (21)

us(s, v, r, t) = 1, s = smax. (22)

For v = vmax, the following Dirichlet condition is prescribed:

u(s, v, r, t) = s, v = vmax. (23)

Remarking that the nodes which are located on the boundary v = 0 are considered as interiornodes and we take a fact into consideration that they must read the PDE model. That is to say,we incorporate the semi-discretized equations at this boundary.

At last, for r = ±rmax, we impose:

ur(s, v, r, t) = 0, r = rmax, (24)

ur(s, v, r, t) = 0, r = −rmax. (25)

By incorporating the above mentioned conditions, we obtain the following systemof semi-discretized ODEs as follows:

U(t) = A(t)U(t), (26)

where A(t) is the coefficient matrix including the boundaries.

Integrator

For discretizing in temporal variable t, many schemes are existing, for example refer to [23].Explicit methods are basically straightforward to implement, but suffer from stability problems.Implicit schemes are unconditionally stable, but only exhibit low convergence or very time-consumingbecause of solving nonlinear system of algebraic equations per step.

Consider uι to be the computational solution for the exact solution U(tι) and choose k+ 1 temporalnodes with the step size Δt = T

k .At the moment, we use the δ–stage Runge–Kutta scheme [23] at tι+1 = tι + Δt, (0 ≤ ι ≤ k) by:

gi = uι + Δtδ

∑j=1

kjai,j,

ki = f (Δtci + tι, gi) ,

uι+1 = uι + Δtδ

∑i=1

biki,

(27)

473


wherein f is defined based on the right hand side of (26). It is generally assumed that the row-sumconditions hold:

ci =δ

∑j=1

ai,j, i = 1, 2, . . . , δ. (28)

Now we consider a sixth-order explicit Runge–Kutta scheme (RK6) below [24]:

Λ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 01 0 0 0 0 0 038

18 0 0 0 0 0

827

227

827 0 0 0 0

3(3p1−7)392

(p1−7)49

−6(p1−7)49

3(p1−21)392 0 0 0

−3(17p1+77)392

(−p1−7)49 − (8p1)

493(121p1+21)

1960(p1+6)

5 0 0(7p1+22)

1223

2(7p1−5)9

−7(3p1−2)20

−7(9p1+49)90

−7(p1−7)18 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (29)

with b = (9/180, 0, 64/180, 0, 49/180, 49/180, 9/180), C = (1, 1/2, 2/3, (7− p1)/14, (7 + p1)/14, 1),and p1 = 211/2.

Notice that a consequence of explicitness is c1 = 0 in (28), so that the function is sampledat the beginning of the current integration step. Here, the sixth-order time-stepping solver consistsof seven stages and reaches sixth order of convergence. The sixth order says that the error of localtruncation is on the order of O(Δt7), while the total accumulated error is on the order of O(Δt6).

In the sequel, we study that under what criteria the numerical discretized solution doesnot blow up. The following theorem is one of the contributions of this work. This is givenfor the time-independent case, i.e., when A(t) = A.

Theorem 2. If the system of ODEs (26) reads the condition of Lipschitz, then the time-stepping method (27)–(29)has conditional stability.

Proof. To find a stability conditions, we proceed as follows. Incorporating the time-stepping solver (27)on the system of ODEs (26) yields:

uι+1 =

(I + ΔtA +

(ΔtA)2

2!

+(ΔtA)3

3!+

(ΔtA)4

4!

+(ΔtA)5

5!+

(ΔtA)6

6!

− (ΔtA)7

2160

)uι.

(30)

Thus, the numerical stability is reduced to:∣∣∣∣1 + Δtωi +(Δtωi)

2

2+

(Δtωi)3

6+

(Δtωi)4

24

+(Δtωi)

5

120+

(Δtωi)6

720− (Δtωi)

7

2126

∣∣∣∣ ≤ 1,(31)

474


which is due to (30) for any ωi as the eigenvalue of A. Now by considering: ωmax (A) , the stabilitycondition can now be represented as follows:∣∣∣∣1 + Δtωmax +

(Δtωmax)2

2+

(Δtωmax)3

6+

(Δtωmax)4

24

+(Δtωmax)5

120+

(Δtωmax)6

720− (Δtωmax)7

2126

∣∣∣∣ ≤ 1.(32)

Noting that the negative semi-definiteness of A makes all its eigenvalues to have negative realparts. Thus, the proposed scheme has numerical stability if the temporal step size Δt satisfy (32).Noting that this can be computed in the language Mathematica [22] via the command

Eigenvalues[matrix, 1]. (33)

The proof is ended.

4. Experiments

In this section, some tests were given for our proposed method showed via Adaptive FiniteDifference Method (AFDM) to price at the money call options, when T = 1 year and E = 100$.A comparison was done by the standard uniform FD scheme [4], which by second order FDapproximations and the Euler’s scheme as a temporal solver shown by FDM. We also comparewith the method provided in [13] shown by Haentjens-In’t Method (HIM).

Mathematica 11.0 is used for the simulations [25]. Time is also reported in second while we employthe following stopping condition:

Error =∣∣∣∣uapprox(s, v, r, t)− uref(s, v, r, t)

uref(s, v, r, t)

∣∣∣∣ , (34)

wherein uref and uapprox are the exact and numerical results.To increase the computational efficiency for very large scale semi-discrete systems that we are

dealing with, here we set AccuracyGoal→ 5, PrecisionGoal→ 5.Here, we consider more number of discretization nodes along s rather than v and r,

since its working interval is larger than the others and the non-smoothness of the initial conditionoccurs along this spatial variable.

The non-constant b is defined as follows:

b(τ) = c1 − c2 exp (−c3τ), τ ≥ 0, (35)

where c1, c2, c3 are constants, and τ = T − t. The following two test cases are considered:

1. κ = 3.0, η = 0.12, a = 0.20, σ1 = 0.80, σ2 = 0.03, ρ12 = 0.6, ρ13 = 0.2, ρ23 = 0.4, c1 = 0.05, c2 = 0,c3 = 0, where the reference value is uref(100, 0.04, 0.1, 1) � 16.176.

2. κ = 0.5, η = 0.8, a = 0.16, σ1 = 0.90, σ2 = 0.03, ρ12 = −0.5, ρ13 = 0.2, ρ23 = 0.1, c1 = 0.055,c2 = 0, c3 = 0, where the reference value is uref(100, 0.04, 0.1, 1) � 20.994.

The results are brought forward in Tables 1 and 2 showing the stable and efficient valuationsof options under HHW PDE via the new high-order procedure. Furthermore, to reveal the positivityand stability of the numerical results, in Experiment 2, and by considering m = 30, n = 18 and o = 18discretization nodes, the results based on AFDM are plotted in Figures 1 and 2.

475


Table 1. Error results in Heston–Hull–White (HHW) Option 1.

Procedure m n o Size Δt Price Error CPU Timing

FDM

10 8 6 480 0.002 25.492 5.7× 10−1 0.3914 10 10 1400 0.001 11.098 3.1× 10−1 0.6218 12 12 2592 0.0005 17.203 6.3× 10−2 1.2224 14 14 4704 0.00025 18.731 1.5× 10−1 3.3928 16 16 7168 0.0002 13.329 1.7× 10−1 6.0745 22 22 21,780 0.00005 14.636 9.4× 10−2 70.54

HIM

10 8 6 480 0.001 14.472 1.0× 10−1 0.4414 10 10 1400 0.0005 15.300 5.3× 10−2 0.8518 12 12 2592 0.00025 15.615 3.4× 10−2 2.0924 14 14 4704 0.0001 15.806 2.2× 10−2 8.0428 16 16 7168 0.0001 15.871 1.8× 10−2 11.8450 22 22 24,200 0.000025 16.006 5.9× 10−3 186.77

AFDM

10 8 6 480 0.005 15.123 6.5× 10−2 0.4114 10 10 1400 0.002 15.986 1.1× 10−2 0.8218 12 12 2592 0.001 16.059 7.2× 10−3 1.9924 14 14 4704 0.000625 16.136 2.4× 10−3 5.5628 16 16 7168 0.0005 16.160 9.8× 10−4 9.0250 22 22 24,200 0.0001 16.179 1.8× 10−4 103.26

Table 2. Error results in HHW option 2.

Procedure m n o Size Δt Price Error CPU Timing

FDM

20 10 10 2000 0.00025 22.022 4.9× 10−2 1.7724 12 12 3456 0.0002 21.436 2.1× 10−2 3.1426 14 14 5096 0.0001 19.678 6.1× 10−2 8.6728 16 16 7168 0.0001 17.376 1.7× 10−1 11.7230 18 18 9720 0.00005 17.404 1.7× 10−1 35.5036 20 20 14,400 0.000025 20.510 2.2× 10−2 107.8938 22 22 18,392 0.000025 20.275 3.3× 10−2 161.1642 22 22 20,328 0.00002 18.370 1.2× 10−1 244.32

HIM

20 10 10 2000 0.00025 20.631 1.6× 10−2 1.6924 12 12 3456 0.0002 20.709 1.2× 10−2 3.4026 14 14 5096 0.0001 20.729 1.1× 10−2 8.6428 16 16 7168 0.0001 20.748 1.0× 10−2 12.3630 18 18 9720 0.00005 20.767 9.9× 10−3 36.5436 20 20 14,400 0.000025 20.810 7.8× 10−3 108.9238 22 22 18,392 0.000025 20.818 7.5× 10−3 166.0842 22 22 20,328 0.00002 20.833 6.7× 10−3 250.67

AFDM

20 10 10 2000 0.0005 20.832 7.7× 10−3 0.8924 12 12 3456 0.0004 20.899 4.5× 10−3 3.3626 14 14 5096 0.00025 20.910 4.0× 10−3 6.6228 16 16 7168 0.0002 20.926 3.2× 10−3 11.2730 18 18 9720 0.0001 20.951 2.0× 10−3 34.5536 20 20 14,400 0.0000625 20.972 1.0× 10−3 98.2238 22 22 18,392 0.00005 20.980 6.6× 10−4 165.2742 22 22 20,328 0.00004 20.999 2.3× 10−4 240.25

476


Figure 1. A numerical solution based on AFDM in Heston–Hull–White (HHW) option 2.

Figure 2. A numerical solution based on AFDM in HHW option 2.

5. Ending Comments

In financial engineering, it is famous that the Black–Scholes PDE could not be useful in realapplication due to several restrictions. Several ideas to observe the market’s reality are models basedupon the stochastic volatility and interest rate models. The resulted PDE problem in this way is hardto be solved theoretically due to higher involved dimensions and so numerical methods are required.

In this paper, we have proposed a new discretized numerical method based on adaptive FDmethodology on non-uniform grids in order to tackle an important problem in computational financeknown as HHW PDE (1). It was proved that the new procedure has conditional stability and shownto be efficient in practice.

477


Further discussions can be investigated to extend the results of this work for other types of optionsdefined on HHW model such as digital (binary) options, at which the initial condition is not onlynon-smooth at the strike but also discontinues.


Acknowledgments: This work was supported by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, Jeddah, under grant No. (D-235-130-1439). The authors, therefore, gratefully acknowledge the DSRtechnical and financial support.

Conflicts of Interest: The author declares no conflict of interest.

References

1. Brigo, D.; Mercurio, F. Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit, 2nd ed.;Springer Finance: Berlin, Germany, 2007.

2. Cakici, N.; Chatterjee, S.; Chen, R.-R. Default risk and cross section of returns. J. Risk Financ. Manag. 2019,12, 95. [CrossRef]

3. Ballestra, L.V.; Cecere, L. A numerical method to estimate the parameters of the CEV model impliedby American option prices: Evidence from NYSE. Chaos Solitons Fractals 2016, 88, 100–106. [CrossRef]

4. Duffy, D.J. Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach; Wiley:Chichester, UK, 2006.

5. Magoulès, F.; Gbikpi-Benissan, G.; Zou, Q. Asynchronous iterations of parareal algorithm for option pricingmodels. Mathematics 2018, 6, 45. [CrossRef]

6. Fouque, J.-P.; Papanicolaou, G.; Sircar, K.R. Derivatives in Financial Markets with Stochastic Volatility;Cambridge Univ. Press: Cambridge, UK, 2000.

7. Heston, S.L. A closed-form solution for options with stochastic volatility with applications to bondand currency options. Rev. Finan. Stud. 1993, 6, 327–343. [CrossRef]

8. Hull, J.; White, A. Using Hull-White interest rate trees. J. Deriv. 1996, 4, 26–36. [CrossRef]9. Schöbel, R.; Zhu, J. Stochastic volatility with an Ornstein-Uhlenbeck process: An extension. Eur. Financ. Rev.

1999, 3, 23–46. [CrossRef]10. Guo, S.; Grzelak, L.A.; Oosterlee, C.W. Analysis of an affine version of the Heston-Hull-White option pricing

partial differential equation. Appl. Numer. Math. 2013, 72, 143–159. [CrossRef]11. Sargolzaei, P.; Soleymani, F. A new finite difference method for numerical solution of Black-Scholes PDE.

Adv. Diff. Equat. Control Process. 2010, 6, 49–55.12. Soleymani, F.; Barfeie, M. Pricing options under stochastic volatility jump model: A stable adaptive scheme.

Appl. Numer. Math. 2019, 145, 69–89. [CrossRef]13. Haentjens, T.; In’t Hout, K.J. Alternating direction implicit finite difference schemes

for the Heston-Hull-White partial differential equation. J. Comput. Fin. 2012, 16, 83–110. [CrossRef]14. Soleymani, F.; Akgül, A. Asset pricing for an affine jump-diffusion model using an FD method of lines

on non-uniform meshes. Math. Meth. Appl. Sci. 2019, 42, 578–591. [CrossRef]15. Itkin, A.; Carr, P. Jumps without tears: A new splitting technology for Barrier options. Int. J. Numer.

Anal. Model. 2011, 8, 667–704.16. Sumei, Z.; Jieqiong, Z. Efficient simulation for pricing barrier options with two-factor stochastic volatility

and stochastic interest rate. Math. Prob. Eng. 2017, 2017, 3912036. [CrossRef]17. Soleymani, F.; Saray, B.N. Pricing the financial Heston-Hull-White model with arbitrary correlation factors

via an adaptive FDM. Comput. Math. Appl. 2019, 77, 1107–1123. [CrossRef]18. Kwok, Y.K. Mathematical Models of Financial Derivatives, 2nd ed.; Springer: Heidelberg, Germany, 2008.19. Ballestra, L.V.; Sgarra, C. he evaluation of American options in a stochastic volatility model with jumps:

An efficient finite element approach. Comput. Math. Appl. 2010, 60, 1571–1590. [CrossRef]20. Fornberg, B. A Practical Guide to Pseudospectral Methods; Cambridge University Press: Cambridge , UK, 1996.21. Soleymani, F.; Barfeie, M.; Khaksar Haghani, F. Inverse multi-quadric RBF for computing the weights of FD

method: Application to American options. Commun. Nonlinear Sci. Numer. Simul. 2018, 64, 74–88. [CrossRef]22. Wellin, P.R.; Gaylord, R.J.; Kamin, S.N. An Introduction to Programming with Mathematica, 3rd ed.;

Cambridge University Press: Cambridge, UK, 2005.

478


23. Sofroniou, M.; Knapp, R. Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica,Tutorial Collection; Wolfram Research, Inc.: Champaign, IL, USA, 2008.

24. Luther, H.A. An explicit sixth-order Runge-Kutta formula. Math. Comput. 1967, 22, 434–436. [CrossRef]25. Mangano, S. Mathematica Cookbook; O’Reilly Media: Sevastopol, CA, USA, 2010.

c© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Iterative Methods for Solving Nonlinear Equations and Systems

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